Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential

Abstract

We consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation $f(z,\cdot )$ is $(p-1)$-sublinear and then the case where it is $(p-1)$-superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter $\lambda \in {\mathbb {R}}$ which we specify exactly in terms of principal eigenvalue of the differential operator.

1 Introduction

Let $\varOmega \subseteq {\mathbb {R}}^N$ be a bounded domain with a $C^2$-boundary $\partial \varOmega$. In this paper we study the following nonlinear parametric Robin problem

In this problem $\varDelta _p$ denotes the p-Laplace differential operator defined by

$$\begin{aligned} \varDelta _p u= \text{ div } (|\nabla u|^{p-2}\nabla u) \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ) \quad (1<p<+\infty ). \end{aligned}$$

Also $\xi (\cdot ) \in L^\infty (\varOmega )$ is an indefinite (that is, sign changing) potential function, $\lambda \in {\mathbb {R}}$ is a parameter and f(z, x) is a Carathéodory perturbation function (that is, for all $x \in {\mathbb {R}}$, $z \rightarrow f(z,x)$ is measurable and for a.a. $z \in \varOmega$, $x \rightarrow f(z,x)$ is continuous). In the boundary condition $\dfrac{\partial u}{\partial n_p}$ denotes the generalized normal derivative defined by

$$\begin{aligned} \dfrac{\partial u}{\partial n_p}=|\nabla u|^{p-2}(\nabla u, n)_{{\mathbb {R}}^N}=|\nabla u|^{p-2} \dfrac{\partial u}{\partial n} \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ), \end{aligned}$$

with $n(\cdot )$ being the outward unit normal on $\partial \varOmega$. This kind of generalized normal derivative is dictated by the nonlinear Green’s identity (see, for example, Gasiński–Papageorgiou [8] (p. 211)). The boundary weight term $\beta \in C^{0,\alpha }(\partial \varOmega )$ ($0<\alpha <1$) and $\beta (z) \ge 0$ for all $z \in \partial \varOmega$.

Problem $(P_{\lambda })$ can be viewed as a perturbation of the usual eigenvalue problem for the Robin p-Laplacian plus an indefinite potential. We look for positive solutions and we consider two distinct cases depending on the growth of the perturbation $f(z,\cdot )$ near $+\infty$:

• $f(z,\cdot )$ is $(p-1)$-sublinear.

• $f(z,\cdot )$ is $(p-1)$-superlinear.

Let ${\widehat{\lambda }}_1 \in {\mathbb {R}}$ be the principal eigenvalue of the differential operator $u \rightarrow - \varDelta _p u + \xi (z)|u|^{p-2}u$ with Robin boundary condition. In the first case ($(p-1)$-sublinear perturbation), we show that for all $\lambda \ge {\widehat{\lambda }}_1$, problem $(P_{\lambda })$ has no positive solution, while for $\lambda < {\widehat{\lambda }}_1$, problem $(P_{\lambda })$ has at least one positive solution. Moreover, this positive solution is unique, if we impose a monotonicity condition on the quotient $x \rightarrow \dfrac{f(z,x)}{x^{p-1}}$ for $x>0$. In the second case ($(p-1)$-superlinear perturbation), the situation changes and uniqueness of the positive solution fails. In fact the problem exhibits a kind of bifurcation phenomenon. Namely, for $\lambda \ge {\widehat{\lambda }}_1$ problem $(P_{\lambda })$ has no positive solution, while for $\lambda < {\widehat{\lambda }}_1$ problem $(P_{\lambda })$ has at least two positive solutions. Finally for both situations, we establish the existence of minimal positive solutions. Our work here extends to the p-Laplacian that of Papageorgiou–Rǎdulescu–Repovš [20]. Eigenvalue problems for the p-Laplacian plus an indefinite potential were studied by Papageorgiou–Rǎdulescu [18] (semilinear problems (that is, $p=2$) with Robin boundary condition) and by Mugnai–Papageorgiou [16] (nonlinear problems with Neumann boundary condition (that is, $\beta \equiv 0$)). Both works deal with nonparametric problems and prove existence and multiplicity results under resonance conditions. We also mention the works of Hu–Papageorgiou [10,11,12]. In [11] the authors treat superdiffusive logistic equation with Robin boundary condition, while in [10, 12], they deal with equations driven by a nonhomogeneous differential operator.

2 Auxiliary results

In this section we present some auxiliary results and notions which we will need in the sequel.

First we deal with the following eigenvalue problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varDelta _p u(z)+ \xi (z)| u(z)|^{p-2}u(z)={\widehat{\lambda }} | u(z)|^{p-2}u(z) &{}\quad \text{ in } \varOmega ,\\ \dfrac{\partial u}{\partial n_p} +\beta (z)| u|^{p-2}u=0 &{}\quad \text{ on } \partial \varOmega .\end{array}\right. } \end{aligned}$$

(1)

Our hypotheses on the functions $\xi (\cdot )$ and $\beta (\cdot )$ are the following:

$H(\xi )$::

$\xi \in L^\infty (\varOmega )$.

$H(\beta )_1$::

$\beta \in C^{0,\alpha }(\partial \varOmega )$ with $\alpha \in (0,1)$ and $\beta (z) \ge 0$ for all $z \in \partial \varOmega$.

In addition to the Sobolev space $W^{1,p}(\varOmega )$, we will also use the Banach space $C^1({\overline{\varOmega }})$ which is an ordered Banach space with positive cone $C_+=\{u \in C^1({\overline{\varOmega }}) \, : \, u(z) \ge 0 \text{ for } \text{ all } z \in {\overline{\varOmega }}\}$. This cone has a nonempty interior given by

$$\begin{aligned} D_+=\left\{ u \in C_+ \, : \, u(z) > 0 \text{ for } \text{ all } z \in {\overline{\varOmega }}\right\} . \end{aligned}$$

Also on $\partial \varOmega$ we consider the $(N-1)$-dimensional Hausdorff (surface) measure $\sigma (\cdot )$. With this measure on $\partial \varOmega$, we can define the Lebesgue spaces $L^\tau (\partial \varOmega )$ $1 \le \tau \le +\infty$. We know that there exists a unique continuous linear map $\gamma _0 : W^{1,p}(\varOmega ) \rightarrow L^p(\partial \varOmega )$ known as the “trace map” s.t. $\gamma _0(u)=u |_{\partial \varOmega }$ for all $u \in W^{1,p}(\varOmega ) \cap C({\overline{\varOmega }})$. So, we understand the trace map as representing the “boundary values” of a Sobolev function $u \in W^{1,p}(\varOmega )$. We know that $\gamma _0$ is compact into $L^\tau (\partial \varOmega )$ for all $\tau \in \left[ 1, \dfrac{(N-1)p}{N-p}\right)$ when $p<N$ and into $L^\tau (\partial \varOmega )$ for all $\tau \in [1,+\infty )$ when $p \ge N$. Moreover, we have

$$\begin{aligned} {\mathrm{im }}\, \gamma _0=W^{\frac{1}{p'},p}(\partial \varOmega ) \quad \left( \frac{1}{p}+\frac{1}{p'}=1\right) \quad \text{ and } \quad \mathrm{ker } \, \gamma _0= W^{1,p}_0(\varOmega ). \end{aligned}$$

In the sequel for the sake of notational simplicity we drop the use of the trace map $\gamma _0$. It is understood that all restrictions of Sobolev functions on $\partial \varOmega$ are taken in the sense of traces.

In what follows by $\vartheta : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ we denote the $C^1$-functional defined by

$$\begin{aligned} \vartheta (u)= \Vert \nabla u\Vert _p^p + \int _\varOmega \xi (z)|u|^pdz + \int _{\partial \varOmega } \beta (z)|u|^p d \sigma \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

From Fragnelli–Mugnai–Papageorgiou [7], we have the following proposition concerning problem (1) (see also Mugnai–Papageorgiou [16] and Papageorgiou–Rǎdulescu [18] where special cases of (1) are investigated).

Proposition 1

If hypotheses $H(\xi )$, $H(\beta )_1$ hold, then problem (1) admits a smallest eigenvalue ${\widehat{\lambda }}_1 \in {\mathbb {R}}$ s.t.

• $$\begin{aligned} {\widehat{\lambda }}_1=\inf \left[ \frac{\vartheta (u)}{\Vert u\Vert _p^p} \, : \, u \in W^{1,p}(\varOmega ), \, u \ne 0 \right] . \end{aligned}$$

  (2)

• ${{\widehat{\lambda }}}_1$ is isolated and simple.

• The infimum in (2) is realized on the one-dimensional eigenspace of ${\widehat{\lambda }}_1$; the elements of this eigenspace do not change sign and if ${\widehat{u}}_1$ denotes the positive, $L^p$-normalized (that is, $\Vert {\widehat{u}}_1\Vert _p=1$) eigenfunction, then ${\widehat{u}}_1 \in D_+$.

  • If ${\widehat{\lambda }}>{\widehat{\lambda }}_1$ is another eigenvalue and ${\widehat{u}} \in W^{1,p}(\varOmega )$ a corresponding eigenfunction, then ${\widehat{u}} \in C^1({\overline{\varOmega }})$ is nodal (that is, sign changing).

As a consequence of these properties, we have the following useful lemma.

Lemma 1

If hypotheses $H(\xi )$, $H(\beta )_1$ hold, $\eta \in L^\infty (\varOmega )$, $\eta (z) \le {\widehat{\lambda }}_1$ for a.a. $z \in \varOmega$ and the inequality is strict on a set of positive measure, then there exists ${\widehat{c}}>0$ s.t.

$$\begin{aligned} {\widehat{c}} \Vert u\Vert ^p \le \vartheta (u)-\int _\varOmega \eta (z)|u|^pdz \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

Proof

Let $\zeta : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ be the $C^1$-functional defined by

$$\begin{aligned} \zeta (u)=\vartheta (u)-\int _\varOmega \eta (z)|u|^pdz \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

From (2) we have $\zeta \ge 0$. Suppose that the claim of the lemma is not true. Then we can find $\{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega )$ s.t.

$$\begin{aligned} \zeta (u_n) \downarrow 0 \quad \text{ as } n \rightarrow +\infty . \end{aligned}$$

(3)

The p-homogeneity of $\zeta (\cdot )$ implies that we may assume that $\Vert u_n\Vert _p=1$ for all $n \in {\mathbb {N}}$. Then clearly $\{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega )$ is bounded (see hypotheses $H(\xi )$, $H(\beta )_1$) and so we may assume that

$$\begin{aligned} u_n \xrightarrow {w} u \text{ in } W^{1,p}(\varOmega ) \quad \text{ and } \quad u_n \rightarrow u \text{ in } L^p(\varOmega ) \text{ and } \text{ in } L^p(\partial \varOmega ), \, \Vert u\Vert _p=1. \end{aligned}$$

(4)

From (3) and (4), we obtain

$$\begin{aligned}&\vartheta (u) \le \int _\varOmega \eta (z)|u|^p dz \le {\widehat{\lambda }}_1 \Vert u\Vert _p^p={\widehat{\lambda }}_1,\nonumber \\&\quad \Rightarrow \vartheta (u)={\widehat{\lambda }}_1 \quad \text{(see } (2)),\nonumber \\&\quad \Rightarrow u= \mu {\widehat{u}}_1 \quad \text{ with } \mu \ne 0~(\text{ see } \text{ Proposition }~1). \end{aligned}$$

(5)

To fix things we assume that $\mu >0$ (the reasoning is the same if $\mu <0$). Then from (5) and since $u=\mu {\widehat{u}}_1 \in D_+$, we have

$$\begin{aligned} \vartheta (u) < {\widehat{\lambda }}_1 \end{aligned}$$

wich contradicts (2). $\square$

Let $A:W^{1,p}(\varOmega ) \rightarrow W^{1,p}(\varOmega )^*$ be the nonlinear map defined by

$$\begin{aligned} \langle A(u),h \rangle =\int _\varOmega |\nabla u |^{p-2}(\nabla u, \nabla h)_{{\mathbb {R}}^N}dz \quad \text{ for } \text{ all } u,h \in W^{1,p}(\varOmega ). \end{aligned}$$

From Motreanu–Motreanu–Papageorgiou [14] (p. 40), we have the following result concerning this map.

Proposition 2

The map $A(\cdot )$ is bounded (that is, maps bounded sets to bounded sets), monotone, continuous (hence maximal monotone too) and of type $(S)_+$, that is, if $u_n \xrightarrow {w} u$ in $W^{1,p}(\varOmega )$ and $\limsup _{n \rightarrow + \infty } \langle A(u_n),u_n-u\rangle \le 0$, then $u_n \rightarrow u$ in $W^{1,p}(\varOmega )$.

Recall that if X is a Banach space and $\varphi \in C^1(X, {\mathbb {R}})$, then we say that $\varphi$ satisfies the Cerami condition (the C-condition for short), if the following is true:

“Every sequence $\{u_n\}_{n \ge 1} \subseteq X$ s.t. $\{\varphi (u_n)\}_{n \ge 1} \subseteq {\mathbb {R}}$ is bounded and $(1 + \Vert u_n\Vert ) \varphi '(u_n) \rightarrow 0$ in $X^*$ as $n \rightarrow +\infty$, admits a strongly convergent subsequence”.

Let $f_0: \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ be a Carathéodory function s.t.

$$\begin{aligned} |f_0(z,x)| \le a(z) (1+|x|^{p^*-1}) \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \in {\mathbb {R}}, \end{aligned}$$

with $a \in L^\infty (\varOmega )_+$ and $p^*= {\left\{ \begin{array}{ll} \frac{Np}{N-p} &{} \text{ if } p<N\\ +\infty &{}\text{ if } N \le p\end{array}\right. }$ (the critical Sobolev exponent). Let $F_0(z,x)= \int _0^x f_0(z,s)ds$ and consider the $C^1$-functional $\varphi _0:W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \varphi _0(u)= \frac{1}{p}\vartheta (u) - \int _\varOmega F_0(z,u)dz \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

From Papageorgiou–Rǎdulescu [17], we have the following result relating local minimizers of $\varphi _0$ and which is an outgrowth of the nonlinear regularity theory. The first such result was proved by Brezis-Nirenberg [4] for $p=2$ and the space $H_0^1(\varOmega )$.

Proposition 3

If $u_0 \in W^{1,p}(\varOmega )$ is a local $C^1({\overline{\varOmega }})$-minimizer of $\varphi _0$, that is, there exists $\delta _1>0$ s.t.

$$\begin{aligned} \varphi _0(u_0) \le \varphi _0(u_0+h) \quad \text{ for } \text{ all } h \in C^1({\overline{\varOmega }}), \Vert h\Vert _{C^1({\overline{\varOmega }})}\le \delta _1, \end{aligned}$$

then $u_0 \in C^{1,\tau }({\overline{\varOmega }})$ with $\tau \in (0,1)$ and it is also a local $W^{1,p}(\varOmega )$-minimizer of $\varphi _0$, that is, there exists $\delta _2>0$ s.t.

$$\begin{aligned} \varphi _0(u_0) \le \varphi _0(u_0+h) \quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ), \Vert h\Vert \le \delta _2. \end{aligned}$$

To make good use of this result, we need a strong comparison principle. In this direction we have the following proposition which is a special case of a more general result due to Fragnelli–Mugnai–Papageorgiou [6]. Given $h_1,h_2 \in L^\infty (\varOmega )$, we say that $h_1 \prec h_2$ if and only if for every $K \subseteq \varOmega$ compact, there exists $\varepsilon =\varepsilon (K)>0$ s.t.

$$\begin{aligned} h_1(z) + \varepsilon \le h_2(z) \quad \text{ for } \text{ a.a. } z \in K. \end{aligned}$$

Note that if $h_1,h_2 \in C(\varOmega )$ and $h_1(z)<h_2(z)$ for all $z \in \varOmega$, then $h_1 \prec h_2$.

Proposition 4

If $\xi , h_1,h_2 \in L^\infty (\varOmega )$, $h_1 \prec h_2$, $u \in C^1({\overline{\varOmega }}) {\setminus } \{0\}$, $v \in D_+$ and they satisfy

$$\begin{aligned}&- \varDelta _p u(z)+\xi (z)|u(z)|^{p-2}u(z)=h_1(z) \quad \text{ for } \text{ a.a. } z \in \varOmega , \\&- \varDelta _p v(z)+\xi (z) v(z)^{p-1}=h_2(z) \quad \text{ for } \text{ a.a. } z \in \varOmega , \,\dfrac{\partial v}{\partial n}<0 \text{ on } \partial \varOmega , \end{aligned}$$

then $(v-u)(z)>0$, for all $z \in \varOmega$ and $\dfrac{\partial (v-u)}{\partial n}\Big |_{D_0}<0$ where $D_0=\{z \in \partial \varOmega : v(z)=u(z)\}$.

Remark 1

If in $C^1({\overline{\varOmega }})$ we introduce the order cone

$$\begin{aligned} {\widehat{C}}_+=\left\{ y \in C^1({\overline{\varOmega }}): y(z) \ge 0 \text{ for } \text{ all } z \in {\overline{\varOmega }}, \, \dfrac{\partial y}{\partial n} \le 0 \text{ on } D_0\right\} \end{aligned}$$

then the above proposition says that $v-u \in {\mathrm{int }} \, {\widehat{C}}_+$. If $D_0 = \emptyset$, then ${\widehat{C}}_+=C_+$.

For problem $(P_{\lambda })$, we introduce the following two sets:

$$\begin{aligned} {\mathcal {L}}&=\{\lambda \in {\mathbb {R}}: \text{ problem } (P_{\lambda }) \text{ admits } \text{ a } \text{ positive } \text{ solution }\},\\ S(\lambda )&=\{ \text{ set } \text{ of } \text{ positive } \text{ solutions } \text{ for } \text{ problem } (P_{\lambda })\}. \end{aligned}$$

For the set $S(\lambda )$ we have the following general result.

Proposition 5

If hypotheses $H(\xi )$, $H(\beta )_1$ hold and $f: \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function s.t. for a.a. $z \in \varOmega$, $f(z,0)=0$, $f(z,x) \ge 0$ for all $x > 0$, $f(z,x)=0$ for all $x < 0$ and $f(z,x) \le a(z)(1+x^{p^*-1})$ for a.a. $z \in \varOmega$, all $x \ge 0$, with $a \in L^\infty (\varOmega )_+$, then $S(\lambda ) \subseteq D_+$ (possibly empty).

Proof

Suppose that $u \in S(\lambda )$. Then

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varDelta _p u(z)+ \xi (z) u(z)^{p-1}=\lambda u(z)^{p-1} +f(z,u(z)) &{}\quad \text{ for } \text{ a.a. } z \in \varOmega ,\\ \dfrac{\partial u}{\partial n_p} +\beta (z) u^{p-1}=0 &{}\quad \text{ on } \partial \varOmega \end{array}\right. } \end{aligned}$$

(6)

(see Papageorgiou–Rǎdulescu [17]). From (6) and Papageorgiou–Rǎdulescu [19] we have $u \in L^\infty (\varOmega )$. Then Theorem 2 of Lieberman [13] implies that $u \in C_+ {\setminus } \{0\}$. From (6) and since $f \ge 0$, we have

$$\begin{aligned}&\varDelta _pu(z) \le (\Vert \xi \Vert _\infty +|\lambda |)u(z)^{p-1} \quad \text{ for } \text{ a.a. } z \in \varOmega , \\&\quad \Rightarrow u \in D_+ \quad \text{(by } \text{ the } \text{ nonlinear } \text{ strong } \text{ maximum } \text{ principle } \text{(see } [8]~\text{(p. } \text{738) }). \end{aligned}$$

$\square$

Proposition 6

If hypotheses $H(\xi )$, $H(\beta )_1$ hold, $f: \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function s.t. for a.a. $z \in \varOmega$ $f(z,0)=0$, $f(z,x)>0$ for all $x>0$, $f(z,x) \le a(z)(1+x^{p^*-1})$ for a.a. $z \in \varOmega$, all $x\ge 0$, with $a \in L^\infty (\varOmega )_+$ and $\lambda \ge {\widehat{\lambda }}_1$, then $S(\lambda )=\emptyset$.

Proof

Arguing by contradiction, suppose that $S(\lambda ) \ne \emptyset$ and let $u \in S(\lambda )$. From Proposition 5 we know that $u \in D_+$. Also, let ${\widehat{u}}_1 \in D_+$ be the principal eigenfunction from Proposition 5. Consider the function

$$\begin{aligned} R({\widehat{u}}_1,u)(z)=|\nabla {\widehat{u}}_1(z)|^p-|\nabla u(z)|^{p-2}\left( \nabla u(z), \nabla \left( \frac{{\widehat{u}}_1^p}{u^{p-1}}\right) (z)\right) _{{\mathbb {R}}^N}. \end{aligned}$$

From the nonlinear Picone’s identity of Allegretto-Huang [2] (see also Motreanu–Motreanu–Papageorgiou [14] (p. 255)), we have

$$\begin{aligned} 0 \le R({\widehat{u}}_1,u)(z) \quad \text{ for } \text{ a.a. } z \in \varOmega . \end{aligned}$$

Then we have

$$\begin{aligned} 0&\le \int _\varOmega R({\widehat{u}}_1,u) dz \\&=\Vert \nabla {\widehat{u}}_1\Vert _p^p - \int _\varOmega |\nabla u|^{p-2}\left( \nabla u, \nabla \left( \frac{{\widehat{u}}_1^p}{u^{p-1}}\right) \right) _{{\mathbb {R}}^N}dz\\&=\Vert \nabla {\widehat{u}}_1\Vert _p^p - \int _\varOmega (-\varDelta _p u)\frac{{\widehat{u}}_1^p}{u^{p-1}} dz + \int _{\partial \varOmega } \beta (z){\widehat{u}}_1^p d \sigma \\&\quad \text {(by the nonlinear Green's identity, see Gasi}\acute{{\text {n}}}\text {ski--Papageorgiou }[8]~\text {(p. 211))}\\&= \Vert \nabla {\widehat{u}}_1\Vert _p^p -\int _\varOmega (\lambda - \xi (z)){\widehat{u}}_1^p dz -\int _\varOmega f(z,u)\frac{{\widehat{u}}_1^p}{u^{p-1}} dz + \int _{\partial \varOmega } \beta (z){\widehat{u}}_1^p d \sigma \\&< \vartheta ({\widehat{u}}_1) - \lambda \quad \text{(since } f(z,u(z))\frac{{\widehat{u}}_1^p}{u^{p-1}}(z)>0 \text{ for } \text{ a.a. } z \in \varOmega \text{ and } \Vert {\widehat{u}}_1\Vert _p=1)\\&= {\widehat{\lambda }}_1 - \lambda \le 0, \end{aligned}$$

a contradiction. Therefore $S(\lambda )=\emptyset$ for all $\lambda \ge {\widehat{\lambda }}_1$. $\square$

3 $(p-1)$-sublinear perturbation

In this section, we deal with the case of a $(p-1)$-sublinear perturbation $f(z,\cdot )$.

$H_1$: $f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function s.t. for a.a. $z \in \varOmega$, $f(z,0)=0$, $f(z,x)>0$ for all $x >0$ and

• (i) for every $\rho >0$, there exists $a_\rho \in L^\infty (\varOmega )_+$ s.t. $f(z,x) \le a_\rho (z)$ for a.a. $z \in \varOmega$, all $x \in [0,\rho ]$;

• (ii) $\lim \nolimits _{x \rightarrow + \infty }\dfrac{f(z,x)}{x^{p-1}}=0$ uniformly for a.a. $z \in \varOmega$;

• (iii) there exist $\delta >0$, $q \in (1,p)$ and $c_1>0$ s.t.

  $$\begin{aligned} c_1x^{q-1} \le f(z,x) \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \in [0,\delta ]. \end{aligned}$$

Remark 2

Since we are looking for positive solutions and the above hypotheses concern the positive semiaxis ${\mathbb {R}}_+=[0,+\infty )$, without any loss of generality we may assume that $f(z,x)=0$ for a.a. $z \in \varOmega$, all $x < 0$. Hypothesis $H_1$(ii) says that for a.a. $z \in \varOmega$ the perturbation $f (z,\cdot )$ is $(p-1)$-sublinear near $+\infty$. Finally hypothesis $H_1$(iii) implies the presence of a concave term near $0^+$.

Example 1

The following functions satisfy hypotheses $H_1$. For the sake of simplicity we drop the z-dependence.

$$\begin{aligned} f_1(x)&=x^{q-1} \quad \text{ for } \text{ all } x \ge 0 \text{ with } 1<q<p,\\ f_2(x)&={\left\{ \begin{array}{ll}x^{q-1}-x^{\tau -1} &{} \text{ if } x \in [0,1],\\ \ln x^{p-1} &{} \text{ if } 1<x, \end{array}\right. } \text{ with } 1<q<p, 1<q< \tau . \end{aligned}$$

Proposition 7

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_1$ hold and $\lambda < {\widehat{\lambda }}_1$, then $S(\lambda ) \ne \emptyset$ and so ${\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)$.

Proof

Let $\eta > \Vert \xi \Vert _\infty$ and consider the following Carathéodory function

$$\begin{aligned} g_\lambda (z,x)={\left\{ \begin{array}{ll}0 &{} \text{ if } x \le 0,\\ (\lambda + \eta )x^{p-1}+f(z,x) &{} \text{ if } 0<x.\end{array}\right. } \end{aligned}$$

(7)

We set $G_\lambda (z,x)=\int _0^x g_\lambda (z,s)ds$ and consider the $C^1$-functional $\varphi _\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \varphi _\lambda (u)=\frac{1}{p}\vartheta (u)+ \frac{\eta }{p}\Vert u\Vert _p^p- \int _\varOmega G_\lambda (z,u)dz \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

Hypotheses $H_1$(i), (ii) imply that given $\varepsilon >0$, we can find $c_2=c_2(\varepsilon )>0$ s.t.

$$\begin{aligned} F(z,x) \le \frac{\varepsilon }{p}x^p +c_2 \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \ge 0. \end{aligned}$$

(8)

Then for all $u \in W^{1,p}(\varOmega )$ we have

$$\begin{aligned} \varphi _\lambda (u)&\ge \frac{1}{p} \vartheta (u) + \frac{\eta }{p}\Vert u^-\Vert _p^p -\frac{\lambda + \varepsilon }{p}\Vert u^+\Vert _p^p- c_2 |\varOmega |_N \quad \text{(see } (7), (8)) \nonumber \\&\ge \frac{1}{p} \vartheta (u) -\frac{\lambda + \varepsilon }{p}\Vert u\Vert _p^p- c_2 |\varOmega |_N . \end{aligned}$$

(9)

Here by $|\cdot |_N$ we denote the Lebesgue measure on ${\mathbb {R}}^N$. Choosing $\varepsilon \in (0, {\widehat{\lambda }}_1 - \lambda )$ (recall that $\lambda < {\widehat{\lambda }}_1$), from (9) and Lemma 1, we have

$$\begin{aligned} \varphi _\lambda (u)&\ge c_3 \Vert u\Vert ^p-c_2|\varOmega |_N \quad \text{ for } \text{ some } c_3>0, \text{ all } u \in W^{1,p}(\varOmega ),\\&\Rightarrow \varphi _\lambda (\cdot ) \text{ is } \text{ coercive }. \end{aligned}$$

Using the Sobolev embedding theorem and the compactness of the trace operator, we see that

$$\begin{aligned} \varphi _\lambda (\cdot ){\text { is sequentially weakly lower semicontinuous.}} \end{aligned}$$

Then invoking the Weierstrass-Tonelli theorem, we can find $u_\lambda \in W^{1,p}(\varOmega )$ s.t.

$$\begin{aligned} \varphi _\lambda (u_\lambda )=\inf \left[ \varphi _\lambda (u) : u \in W^{1,p}(\varOmega ) \right] . \end{aligned}$$

(10)

Let $t \in (0,1)$ be small s.t.

$$\begin{aligned} t {\widehat{u}}_1(z) \in (0,\delta ] \quad \text{ for } \text{ all } z \in {\overline{\varOmega }}\quad \text{(recall } {\widehat{u}}_1 \in D_+). \end{aligned}$$

Here $\delta >0$ is as in hypothesis $H_1$(iii). Then we have

$$\begin{aligned} \varphi _\lambda (t {\widehat{u}}_1)&\le \frac{t^p}{p} \vartheta ({\widehat{u}}_1)- \frac{t^p}{p} \lambda - \frac{t^q}{q} c_1 \Vert {\widehat{u}}_1\Vert _q^q \quad \text{(see } (7) \text{ and } \text{ hypothesis } H_1\text{(iii)) }\\&= \frac{t^p}{p}[ {\widehat{\lambda }}_1 - \lambda ] - \frac{t^q}{q} c_1 \Vert {\widehat{u}}_1\Vert _q^q \quad \text{(see } \text{ Proposition }~1 \text{ and } \text{ recall } \text{ that } \Vert {\widehat{u}}_1\Vert _p=1)\\&= \frac{t^p}{p}c_4 - \frac{t^q}{q} c_5 \quad \text{ with } c_4={\widehat{\lambda }}_1 - \lambda>0, c_5=c_1 \Vert {\widehat{u}}_1\Vert _q^q>0. \end{aligned}$$

Since $t \in (0,1)$ and $q<p$, by choosing $t \in (0,1)$ even smaller if necessary, we have

$$\begin{aligned}&\varphi _\lambda (t {\widehat{u}}_1)<0,\\&\quad \Rightarrow \varphi _\lambda (u_\lambda )<0=\varphi _\lambda (0) \quad \text{(see } (10)),\\&\quad \Rightarrow u_\lambda \ne 0. \end{aligned}$$

From (10), we have

$$\begin{aligned}&\varphi '_\lambda (u_\lambda )=0, \nonumber \\&\quad \Rightarrow \langle A(u_\lambda ),h \rangle + \int _\varOmega (\xi (z)+\eta ) |u_\lambda |^{p-2}u_\lambda h dz + \int _{\partial \varOmega } \beta (z)|u_\lambda |^{p-2}u_\lambda h d \sigma \nonumber \\&\quad = \int _\varOmega [(\lambda +\eta )(u_\lambda ^+)^{p-1}+f(z,u_\lambda ^+)]h dz \quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ). \end{aligned}$$

(11)

In (11) we choose $h=-u_\lambda ^- \in W^{1,p}(\varOmega )$. Then

$$\begin{aligned}&\vartheta (u_\lambda ^-) + \eta \Vert u_\lambda ^-\Vert _p^p=0,\\&\quad \Rightarrow c_6 \Vert u_\lambda ^-\Vert ^p \le 0 \text{ for } \text{ some } c_6>0\\&\qquad \text{(recall } \text{ that } \eta >\Vert \xi \Vert _\infty \text{ and } \text{ see } \text{ hypothesis } H(\beta )_1)\\&\quad \Rightarrow u_\lambda \ge 0, \, u_\lambda \ne 0. \end{aligned}$$

Then equation (11) becomes

$$\begin{aligned}&\langle A(u_\lambda ),h \rangle + \int _\varOmega \xi (z)u_\lambda ^{p-1}h dz + \int _{\partial \varOmega }\beta (z)u_\lambda ^{p-1}h d \sigma \\&\quad =\int _\varOmega [\lambda u_\lambda ^{p-1}+f(z,u_\lambda )]h dz \quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ),\\&\quad \Rightarrow -\varDelta _p u_\lambda (z)+\xi (z)u_\lambda (z)^{p-1}=\lambda u_\lambda (z)^{p-1}+f(z,u_\lambda (z)) \quad \text{ for } \text{ a.a. } z \in \varOmega ,\\&\qquad \dfrac{\partial u_\lambda }{\partial n_p} + \beta (z)u_\lambda ^{p-1}=0 \quad \text{ on } \partial \varOmega ,\\&\quad \Rightarrow u_\lambda \in S(\lambda ) \subseteq D_+ \quad \text{(see } \text{ Proposition }~5 \text{ and } \text{ so } {\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1). \end{aligned}$$

$\square$

In fact we can show that problem $(P_{\lambda })$ for $\lambda < {\widehat{\lambda }}_1$ has a smallest positive solution.

Fix $\lambda < {\widehat{\lambda }}_1$ and $r \in (p,p^*)$. Hypotheses $H_1$(i), (ii), (iii) imply that we can find $c_7(\lambda )>0$ with $\lambda \rightarrow c_7(\lambda )$ bounded on bounded subsets of ${\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)$ s.t.

$$\begin{aligned} \lambda x^{p-1}+f(z,x) \ge c_1 x^{q-1}-c_7(\lambda )x^{r-1} \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \ge 0. \end{aligned}$$

(12)

This unilateral growth estimate for the reaction term of problem $(P_{\lambda })$ leads to the following auxiliary nonlinear Robin problem:

For this problem we have the following existence and uniqueness result.

Proposition 8

If hypotheses $H(\xi )$, $H(\beta )_1$ hold, then for every $\lambda \in {\mathbb {R}}$ problem ($Au_\lambda$) admits a unique positive solution $u_*^\lambda \in D_+.$

Proof

First we show the existence of a positive solution for problem ($Au_\lambda$). To this end, we consider the $C^1$-functional $\psi _\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \psi _\lambda (u)&= \frac{1}{p}\vartheta (u) + \frac{\eta }{p}\Vert u^-\Vert _p^p -\frac{c_1}{q}\Vert u^+\Vert _q^q+\frac{c_7(\lambda )}{r}\Vert u^+\Vert ^r_r \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega )\nonumber \\&\quad \ge \frac{1}{p}\left[ \vartheta (u^-)+\eta \Vert u^-\Vert _p^p\right] + \frac{1}{p}\vartheta (u^+)+\frac{c_7(\lambda )}{r}\Vert u^+\Vert ^r_r -\frac{c_1}{q}\Vert u^+\Vert _q^q . \end{aligned}$$

(13)

We have

$$\begin{aligned}&\frac{1}{p}\vartheta (u^+)+\frac{c_7(\lambda )}{r}\Vert u^+\Vert ^r_r -\frac{c_1}{q}\Vert u^+\Vert _q^q \nonumber \\&\quad \ge \frac{1}{p}\Vert \nabla u^+\Vert _p^p+c_8(\lambda )\Vert u^+\Vert ^r_p -\frac{1}{p}\Vert \xi \Vert _\infty \Vert u^+\Vert _p^p-c_9\Vert u^+\Vert _p^q \quad (\text{ for } \text{ some } c_8(\lambda ),c_9 >0)\nonumber \\&\quad =\frac{1}{p}\Vert \nabla u^+\Vert _p^p+\left[ c_8(\lambda )\Vert u^+\Vert ^{r-p}_p -\frac{1}{p}\Vert \xi \Vert _\infty -\dfrac{c_9}{\Vert u^+\Vert ^{p-q}_p}\right] \Vert u^+\Vert _p^p. \end{aligned}$$

(14)

Using (14) in (13) and recalling that $q<p<r$, we infer that $\psi _\lambda (\cdot )$ is coercive. Also, it is sequentially weakly lower semicontinuous (use the Sobolev embedding theorem and the compactness of the trace map). So, by the Weierstrass-Tonelli theorem, we can find $u_*^\lambda \in W^{1,p}(\varOmega )$ s.t.

$$\begin{aligned} \psi _\lambda (u^\lambda _*)=\inf \left[ \psi _\lambda (u) : u \in W^{1,p}(\varOmega ) \right] . \end{aligned}$$

(15)

Since $q<p<r$, as before (see the proof of Proposition 7), we can show that

$$\begin{aligned}&\psi _\lambda (u^\lambda _*)<0,\\&\quad \Rightarrow u^\lambda _* \ne 0. \end{aligned}$$

From (15) we have

$$\begin{aligned}&\psi '_\lambda (u^\lambda _*)=0, \nonumber \\&\quad \Rightarrow \langle A(u^\lambda _*),h \rangle + \int _\varOmega \xi (z) |u^\lambda _*|^{p-2}u^\lambda _* h dz + \int _{\partial \varOmega } \beta (z)|u^\lambda _*|^{p-2}u^\lambda _* h d \sigma \nonumber \\&\qquad - \eta \int _\varOmega (u^{\lambda ^-}_*)^{p-1}hdz\nonumber \\&\quad =c_1 \int _\varOmega (u^{\lambda ^+}_*)^{q-1} h dz - c_7(\lambda ) \int _\varOmega (u^{\lambda ^+}_*)^{r-1} h dz\quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ). \end{aligned}$$

(16)

In (16) we choose $h=-u^{\lambda ^-}_*\in W^{1,p}(\varOmega )$. Then

$$\begin{aligned}&\vartheta (u^{\lambda ^-}_*) + \eta \Vert u^{\lambda ^-}_*\Vert _p^p=0,\\&\quad \Rightarrow c_{10} \Vert u^{\lambda ^-}_*\Vert ^p \le 0 \text{ for } \text{ some } c_{10}>0~\text{(recall } \text{ that } \eta >\Vert \xi \Vert _\infty ),\\&\quad \Rightarrow u^{\lambda }_* \ge 0, \, u^{\lambda }_* \ne 0. \end{aligned}$$

Therefore Eq. (16) becomes

$$\begin{aligned}&\langle A(u^{\lambda }_*),h \rangle + \int _\varOmega \xi (z)(u^{\lambda }_*)^{p-1}h dz + \int _{\partial \varOmega }\beta (z)(u^{\lambda }_*)^{p-1}h d \sigma \nonumber \\&\quad =c_1 \int _\varOmega (u^{\lambda }_*)^{q-1} h dz - c_7(\lambda ) \int _\varOmega (u^{\lambda }_*)^{r-1} h dz \quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ),\nonumber \\&\quad \Rightarrow -\varDelta _p u^{\lambda }_*(z)+\xi (z)u^{\lambda }_*(z)^{p-1}=c_1 u^{\lambda }_*(z)^{q-1}- c_7(\lambda )u^{\lambda }_*(z)^{r-1} \quad \text{ for } \text{ a.a. } z \in \varOmega ,\nonumber \\&\qquad \dfrac{\partial u^{\lambda }_*}{\partial n_p} + \beta (z)(u^{\lambda }_*)^{p-1}=0 \quad \text{ on } \partial \varOmega \quad \text {(see Papageorgiou-R}{\check{\text {a}}}\text {dulescu }[17]), \\&\quad \Rightarrow u_*^\lambda \text{ is } \text{ a } \text{ positive } \text{ solution } \text{ of } (Au_{\lambda }). \nonumber \end{aligned}$$

(17)

As before, the nonlinear regularity theory (see [13]) implies $u_*^\lambda \in C_+ {\setminus } \{0\}$.

Moreover, from (17) we have

$$\begin{aligned} \varDelta _p u_*^\lambda (z)&\le (c_7(\lambda ) \Vert u_*^\lambda \Vert _\infty ^{r-p}+\Vert \xi \Vert _\infty ) u_*^\lambda (z)^{p-1}\\&\le c_{11}u_*^\lambda (z)^{p-1} \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ some } c_{11}>0,\\&\Rightarrow u^\lambda _* \in D_+ ~\text{(by } \text{ the } \text{ nonlinear } \text{ strong } \text{ maximum } \text{ principle, } [8]~\text{(p. } \text{738)) }. \end{aligned}$$

Next we show the uniqueness of this positive solution. To this end suppose that $v_*^\lambda \in W^{1,p}(\varOmega )$ is another positive solution of ($Au_\lambda$). As above we can show that $v^\lambda _* \in D_+$.

We have

$$\begin{aligned}&\int _\varOmega \left( \frac{c_1}{(u^\lambda _*)^{p-q}}-c_7(\lambda )(u^\lambda _*)^{r-p} \right) \left( (u^\lambda _*)^{p}-(v^\lambda _*)^{p} \right) dz\nonumber \\&\quad =\int _\varOmega \left( c_1(u^\lambda _*)^{q-1}-c_7(\lambda )(u^\lambda _*)^{r-1}\right) \left( u^\lambda _* - \dfrac{(v^\lambda _*)^{p}}{(u^\lambda _*)^{p-1}}\right) dz\nonumber \\&\quad =\int _\varOmega \left( -\varDelta _p u_*^\lambda +\xi (z) (u^\lambda _*)^{p-1}\right) \left( u^\lambda _* - \dfrac{(v^\lambda _*)^{p}}{(u^\lambda _*)^{p-1}}\right) dz \quad \text{(see } (17))\nonumber \\&\quad = \int _\varOmega |\nabla u_*^\lambda |^{p-2} \left( \nabla u_*^\lambda , \nabla \left( u^\lambda _* - \dfrac{(v^\lambda _*)^{p}}{(u^\lambda _*)^{p-1}}\right) \right) _{{\mathbb {R}}^N}dz\nonumber \\&\qquad + \int _\varOmega \xi (z) (u^\lambda _*)^{p-1}\left( u^\lambda _* - \dfrac{(v^\lambda _*)^{p}}{(u^\lambda _*)^{p-1}}\right) dz\nonumber \\&\qquad + \int _\varOmega \beta (z)(u_*^\lambda )^{p-1} \left( u^\lambda _* - \dfrac{(v^\lambda _*)^{p}}{(u^\lambda _*)^{p-1}}\right) d\sigma \nonumber \\&\qquad \text { (using the nonlinear Green's identity, see }[8]~\text{(p. } \text{211) })\nonumber \\&\quad = \Vert \nabla u_*^\lambda \Vert ^p_p -\Vert \nabla v_*^\lambda \Vert ^p_p + \int _\varOmega R(v_*^\lambda , u_*^\lambda )dz + \int _\varOmega \xi (z)\left( (u_*^\lambda )^p - (v_*^\lambda )^p\right) dz \nonumber \\&\qquad + \int _\varOmega \beta (z)\left( (u_*^\lambda )^p - (v_*^\lambda )^p\right) d \sigma . \end{aligned}$$

(18)

Interchanging the roles of $u_*^\lambda$ and $v_*^\lambda$ in the above argument, we also have

$$\begin{aligned}&\int _\varOmega \left( \frac{c_1}{(v^\lambda _*)^{p-q}}-c_7(\lambda )(v^\lambda _*)^{r-p} \right) \left( (v^\lambda _*)^{p}-(u^\lambda _*)^{p} \right) dz \nonumber \\&\quad = \Vert \nabla v_*^\lambda \Vert ^p_p -\Vert \nabla u_*^\lambda \Vert ^p_p + \int _\varOmega R(u_*^\lambda , v_*^\lambda )dz + \int _\varOmega \xi (z)\left( (v_*^\lambda )^p - (u_*^\lambda )^p\right) dz \nonumber \\&\qquad + \int _\varOmega \beta (z)\left( (v_*^\lambda )^p - (u_*^\lambda )^p\right) d \sigma . \end{aligned}$$

(19)

Adding (18) and (19) and using the nonlinear Picone’s identity, we have

$$\begin{aligned} 0&\le \int _\varOmega \left( R(v_*^\lambda , u_*^\lambda ) +R(u_*^\lambda , v_*^\lambda )\right) dz \nonumber \\&= \int _\varOmega \left( c_1\left( \frac{1}{(u^\lambda _*)^{p-q}} -\frac{1}{(v^\lambda _*)^{p-q}}\right) -c_7(\lambda )\left( (u^\lambda _*)^{r-p} -(v^\lambda _*)^{r-p}\right) \right) \left( (u^\lambda _*)^{p}-(v^\lambda _*)^{p} \right) dz. \end{aligned}$$

(20)

Since the function $x \rightarrow \dfrac{c_1}{x^{p-q}}-c_7(\lambda )x^{r-p}$ is strictly decreasing on $(0,+\infty )$, from (20) we infer that

$$\begin{aligned} u_*^\lambda = v_*^\lambda . \end{aligned}$$

This proves the uniqueness of the positive solution $u_*^\lambda \in D_+$ of problem ($Au_\lambda$). $\square$

Remark 3

There is an alternative approach to the uniqueness of the positive solution $u_*^\lambda \in D_+$ of problem ($Au_\lambda$) which does not use the nonlinear Picone’s identity. For this we need to assume that $\beta (z)>0$ for all $z \in \partial \varOmega$. First note that, if $\rho =\Vert u_*^\lambda \Vert _\infty$, then we can find ${\widehat{\xi }}_\rho >0$ s.t. for a.a. $z \in \varOmega$, the function $x \rightarrow c_1 x^{q-1}-c_7(\lambda )x^{r-1}+ {\widehat{\xi }}_\rho x^{p-1}$ is nondecreasing on $[0,\rho ]$. As before let $v_*^\lambda \in D_+$ be another positive solution of ($Au_\lambda$) and let $t>0$ be the biggest real s.t.

$$\begin{aligned} tv_*^\lambda \le u_*^\lambda . \end{aligned}$$

(21)

We assume that $t \in (0,1)$. We have

$$\begin{aligned}&-\varDelta _p (tv_*^\lambda )+(\xi (z)+{\widehat{\xi }}_\rho )(tv_*^\lambda )^{p-1}\\&\quad =t^{p-1}[-\varDelta _p v_*^\lambda +(\xi (z)+{\widehat{\xi }}_\rho )(v_*^\lambda )^{p-1}]\\&\quad =t^{p-1}[c_1 (v_*^\lambda )^{q-1} -c_7(\lambda )(v_*^\lambda )^{r-1}+{\widehat{\xi }}_\rho (v_*^\lambda )^{p-1}]\\&\quad< c_1(tv_*^\lambda )^{q-1}-c_7(\lambda )(tv_*^\lambda )^{r-1} +{\widehat{\xi }}_\rho (tv_*^\lambda )^{p-1} \quad \text{(since } t \in (0,1) \text{ and } q<p<r)\\&\quad \le c_1(u_*^\lambda )^{q-1}-c_7(\lambda )(u_*^\lambda )^{r-1} +{\widehat{\xi }}_\rho (u_*^\lambda )^{p-1} \quad \text{(see } (21))\\&\quad =-\varDelta _p u_*^\lambda +(\xi (z)+{\widehat{\xi }}_\rho )(u_*^\lambda )^{p-1} \quad \text{ for } \text{ a.a. } z\in \varOmega . \end{aligned}$$

Invoking Proposition 4 (recall $\beta >0$), we have

$$\begin{aligned} u_*^\lambda -tv_*^\lambda \in D_+, \end{aligned}$$

(22)

where we recall that ${\widehat{C}}_+=\left\{ y \in C^1({\overline{\varOmega }}): y(z) \ge 0 \text{ for } \text{ all } z \in {\overline{\varOmega }}, \, \dfrac{\partial u}{\partial n}\Big |_{\partial \varOmega }\le 0 \right\}$.

Evidently (22) contradicts the maximality of $t>0$. Therefore we must have $t \ge 1$ and so

$$\begin{aligned} v_*^\lambda \le u_*^\lambda \quad \text{(see } (21)). \end{aligned}$$

Interchanging the roles of $u_*^\lambda \in D_+$ and $v_*^\lambda \in D_+$ in the above argument we also have

$$\begin{aligned}&u_*^\lambda \le v_*^\lambda ,\\&\quad \Rightarrow u_*^\lambda =v_*^\lambda . \end{aligned}$$

So, again we have proved uniqueness of the positive solution of problem ($Au_\lambda$). Recall that $\lambda \rightarrow c_7(\lambda )$ is bounded on bounded sets of $\lambda \in {\mathbb {R}}$. So, if $B \subseteq {\mathbb {R}}$ is bounded, ${\widehat{c}}_7 \ge c_7(\lambda )$ for all $\lambda \in B$ and ${\widehat{u}} \in D_+$ is the unique positive solution of the auxiliary problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varDelta _p u(z)+ \xi (z)u(z)^{p-1}=c_1 u(z)^{q-1}-{\widehat{c}}_7 u(z)^{r-1} &{}\quad \text{ in } \varOmega ,\\ \dfrac{\partial u}{\partial n_p} +\beta (z)u^{p-1}=0 &{}\quad \text{ on } \partial \varOmega , \end{array}\right. } \end{aligned}$$

(see Proposition 8), then ${\widehat{u}} \le u_*^\lambda$ for all $\lambda \in B$.

Next using $u_*^\lambda \in D_+$, we can have a lower bound for the elements of the set $S(\lambda )$. This fact will be used to produce the smallest positive solution for problem $(P_{\lambda })$ when $\lambda < {\widehat{\lambda }}_1$.

So, we have the following result.

Proposition 9

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_1$ hold and $\lambda < {\widehat{\lambda }}_1$, then $u_*^\lambda \le u$ for all $u \in S(\lambda )$.

Proof

As before let $\eta > \Vert \xi \Vert _\infty$. For $u \in S(\lambda )$ we consider the following Carathéodory function

$$\begin{aligned} {\widehat{g}}_\lambda (z,x)={\left\{ \begin{array}{ll} 0 &{}\quad \text{ if } x<0,\\ c_1 x^{q-1}-c_7(\lambda )x^{r-1}+\eta x^{p-1} &{}\quad \text{ if } 0 \le x \le u(z),\\ c_1 u(z)^{q-1}-c_7(\lambda )u(z)^{r-1}+\eta u(z)^{p-1} &{}\quad \text{ if } u(z)<x. \end{array}\right. } \end{aligned}$$

(23)

We set ${\widehat{G}}_\lambda (z,x)= \int _0^x {\widehat{g}}_\lambda (z,s)ds$ and consider the $C^1$-functional ${\widehat{\psi }}_\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\widehat{\psi }}_\lambda (u)= \frac{1}{p}\vartheta (u) + \frac{\eta }{p}\Vert u\Vert _p^p - \int _\varOmega {\widehat{G}}_\lambda (z,u)dz \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

From (23) and since $\eta > \Vert \xi \Vert _\infty$, we see that the functional ${\widehat{\psi }}_\lambda$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find ${\widehat{u}}_*^\lambda \in W^{1,p}(\varOmega )$ s.t.

$$\begin{aligned} {\widehat{\psi }}_\lambda ({\widehat{u}}^\lambda _*)=\inf \left[ {\widehat{\psi }}_\lambda (u) : u \in W^{1,p}(\varOmega ) \right] . \end{aligned}$$

(24)

As before, since $q<p<r$, we have that

$$\begin{aligned}&{\widehat{\psi }}_\lambda ({\widehat{u}}^\lambda _*)<0={\widehat{\psi }}_\lambda (0),\\&\quad \Rightarrow {\widehat{u}}^\lambda _* \ne 0. \end{aligned}$$

From (24) we have

$$\begin{aligned}&{\widehat{\psi }}'_\lambda ({\widehat{u}}^\lambda _*)=0, \nonumber \\&\quad \Rightarrow \langle A({\widehat{u}}^\lambda _*),h \rangle + \int _\varOmega (\xi (z)+\eta ) |{\widehat{u}}^\lambda _*|^{p-2}{\widehat{u}}^\lambda _* h dz + \int _{\partial \varOmega } \beta (z)|{\widehat{u}}^\lambda _*|^{p-2}{\widehat{u}}^\lambda _* h d \sigma \nonumber \\&\quad = \int _\varOmega g_\lambda (z,{\widehat{u}}^\lambda _*) h dz\quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ). \end{aligned}$$

(25)

In (25) first we choose $h=-{\widehat{u}}^{\lambda ^-}_* \in W^{1,p}(\varOmega )$. We obtain

$$\begin{aligned}&\vartheta ({\widehat{u}}^{\lambda ^-}_*) + \eta \Vert {\widehat{u}}^{\lambda ^-}_*\Vert _p^p=0 \quad \text{(see } (23)),\\&\quad \Rightarrow c_{12} \Vert {\widehat{u}}^{\lambda ^-}_*\Vert ^p \le 0 \quad \text{ for } \text{ some } c_{12}>0~\text{(recall } \text{ that } \eta >\Vert \xi \Vert _\infty ),\\&\quad \Rightarrow {\widehat{u}}^{\lambda }_* \ge 0, \, {\widehat{u}}^{\lambda }_* \ne 0. \end{aligned}$$

Next in (25) we choose $({\widehat{u}}^\lambda _*-u)^+ \in W^{1,p}(\varOmega )$. We have

$$\begin{aligned}&\langle A({\widehat{u}}^{\lambda }_*), ({\widehat{u}}^{\lambda }_*-u)^+\rangle + \int _\varOmega (\xi (z)+\eta ) ({\widehat{u}}^{\lambda }_*)^{p-1}({\widehat{u}}^{\lambda }_*-u)^+dz \\&\qquad + \int _{\partial \varOmega } \beta (z) ({\widehat{u}}^{\lambda }_*)^{p-1}({\widehat{u}}^{\lambda }_*-u)^+d \sigma \\&\quad = \int _\varOmega (c_1 u^{q-1} -c_7(\lambda ) u^{r-1}+\eta u^{p-1})({\widehat{u}}^{\lambda }_*-u)^+ dz \quad \text{(see } (23))\\&\quad \le \int _\varOmega (\lambda u^{p-1} +f(z,u)+\eta u^{p-1})({\widehat{u}}^{\lambda }_*-u)^+ dz \quad \text{(see } (12))\\&\quad = \langle A(u), ({\widehat{u}}^{\lambda }_*-u)^+\rangle + \int _\varOmega (\xi (z)+\eta ) u^{p-1}({\widehat{u}}^{\lambda }_*-u)^+dz \\&\qquad + \int _{\partial \varOmega } \beta (z) u^{p-1}({\widehat{u}}^{\lambda }_*-u)^+d \sigma ~\text{(recall } \text{ that } u \in S(\lambda )),\\&\quad \Rightarrow \, \langle A({\widehat{u}}^{\lambda }_*)-A(u), ({\widehat{u}}^{\lambda }_*-u)^+\rangle + \int _\varOmega (\xi (z)+\eta ) (({\widehat{u}}^{\lambda }_*)^{p-1}-u^{p-1})({\widehat{u}}^{\lambda }_*-u)^+dz\\&\qquad + \int _{\partial \varOmega } \beta (z) (({\widehat{u}}^{\lambda }_*)^{p-1}-u^{p-1})({\widehat{u}}^{\lambda }_*-u)^+d \sigma \le 0,\\&\quad \Rightarrow \,{\widehat{u}}^{\lambda }_* \le u \quad \text{(since } \eta > \Vert \xi \Vert _\infty \text{ and } \beta \ge 0, \text{ see } \text{ hypothesis } H(\beta )_1). \end{aligned}$$

Therefore, we have proved that

$$\begin{aligned}&{\widehat{u}}^{\lambda }_* \in [0,u]=\{v \in W^{1,p}(\varOmega ): 0 \le v(z) \le u(z) \text{ for } \text{ a.a. } z\in \varOmega \}, \, {\widehat{u}}^{\lambda }_* \ne 0,\\&\quad \Rightarrow {\widehat{u}}^{\lambda }_* \text{ is } \text{ a } \text{ positive } \text{ solution } \text{ of } (Au_{\lambda })~\text{(see } (25) \text{ and } (23)),\\&\quad \Rightarrow {\widehat{u}}^{\lambda }_*=u^{\lambda }_* \in D_+ \quad \text{(see } \text{ Proposition }~8). \end{aligned}$$

Finally we have

$$\begin{aligned} u^{\lambda }_* \le u \quad \text{ for } \text{ all } u \in S(\lambda ). \end{aligned}$$

$\square$

Proposition 10

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_1$ hold and $\lambda < {\widehat{\lambda }}_1$, then problem $(P_{\lambda })$ admits a smallest positive solution ${\overline{u}}_\lambda \in D_+$.

Proof

As in Filippakis–Papageorgiou [5], we have that $S(\lambda )$ is downward directed, that is, if $u_1, u_2 \in S(\lambda )$, there is $u \in S(\lambda )$ s.t. $u \le u_1$, $u \le u_2$. Invoking Lemma 3.10 of Hu–Papageorgiou [9] (p. 178), we can find $\{u_n\}_{n \ge 1} \subseteq S(\lambda )$ decreasing s.t.

$$\begin{aligned} \inf S(\lambda )=\inf _{n \ge 1}u_n. \end{aligned}$$

We have

$$\begin{aligned} \langle A(u_n),h\rangle + \int _\varOmega \xi (z)u_n^{p-1}h dz + \int _{\partial \varOmega }\beta (z) u_n^{p-1}h d \sigma = \int _\varOmega (\lambda u_n^{p-1}+f(z,u_n))dz\nonumber \\ \end{aligned}$$

(26)

for all $h \in W^{1,p}(\varOmega )$. Since $u_n \le u_1 \in S(\lambda ) \subseteq D_+$, from (26) and hypotheses $H(\xi )$, $H(\beta )_1$, $H_1$(i) it follows that

$$\begin{aligned} \{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega ) \text{ is } \text{ bounded. } \end{aligned}$$

So, we may assume that

$$\begin{aligned} u_n \xrightarrow {w} {\overline{u}}_\lambda \text{ in } W^{1,p}(\varOmega ) \quad \text{ and } \quad u_n \rightarrow {\overline{u}}_\lambda \text{ in } L^{p}(\varOmega ) \text{ and } \text{ in } L^{p}(\partial \varOmega ). \end{aligned}$$

(27)

In (26) we choose $h=u_n - {\overline{u}}_\lambda \in W^{1,p}(\varOmega )$, pass to the limit as $n \rightarrow +\infty$ and use (27). Then we have

$$\begin{aligned}&\lim _{n \rightarrow +\infty }\langle A(u_n),u_n - {\overline{u}}_\lambda \rangle =0,\nonumber \\&\quad \Rightarrow u_n \rightarrow {\overline{u}}_\lambda \quad \text{ in } W^{1,p}(\varOmega )~\text{(see } \text{ Proposition }~2). \end{aligned}$$

(28)

If in (26) we pass to the limit as $n \rightarrow +\infty$ and use (28), then

$$\begin{aligned}&\langle A({\overline{u}}_\lambda ),h \rangle +\int _\varOmega \xi (z){\overline{u}}_\lambda ^{p-1}h dz + \int _{\partial \varOmega } \beta (z){\overline{u}}_\lambda ^{p-1}h d \sigma = \int _\varOmega (\lambda {\overline{u}}_\lambda ^{p-1}+f(z,{\overline{u}}_\lambda ))h dz\\&\quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ),\\&\quad \Rightarrow {\overline{u}}_\lambda \ge 0 \text{ is } \text{ a } \text{ solution } \text{ of } \text{ problem } (P_{\lambda }). \end{aligned}$$

From Proposition 9 we have

$$\begin{aligned}&u_*^\lambda \le u_n \quad \text{ for } \text{ all } n \in {\mathbb {N}},\\&\quad \Rightarrow u_*^\lambda \le {\overline{u}}_\lambda \quad \text{(see } (28)). \end{aligned}$$

Hence ${\overline{u}}_\lambda \ne 0$ and so we conclude that

$$\begin{aligned} {\overline{u}}_\lambda \in S(\lambda ) \subseteq D_+ \quad \text{ and } \quad {\overline{u}}_\lambda =\inf S(\lambda ). \nonumber \\ \end{aligned}$$

$\square$

Next we examine the map $\lambda \rightarrow {\overline{u}}_\lambda$ from $(-\infty , {\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$.

Proposition 11

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_1$ hold, then the map $\lambda \rightarrow {\overline{u}}_\lambda$ from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ is nondecreasing (that is, if $\lambda < \mu$, then ${\overline{u}}_\lambda \le {\overline{u}}_\mu$) and left continuous.

Proof

Suppose that $\lambda , \mu \in {\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ and $\lambda < \mu$. Let ${\overline{u}}_\mu \in S(\mu )$ be the minimal positive solution of problem $(P_\mu )$ (see Proposition 10). For $\eta > \Vert \xi \Vert _\infty$ we introduce the following Carathéodory function

$$\begin{aligned} e_\lambda (z,x)={\left\{ \begin{array}{ll} 0 &{} \quad \text{ if } x<0,\\ (\lambda + \eta ) x^{p-1}+f(z,x) &{}\quad \text{ if } 0 \le x \le {\overline{u}}_\mu (z),\\ (\lambda + \eta ) {\overline{u}}_\mu (z)^{p-1}+f(z,{\overline{u}}_\mu (z)) &{} \quad \text{ if } {\overline{u}}_\mu (z)<x. \end{array}\right. } \end{aligned}$$

(29)

We set $E_\lambda (z,x)= \int _0^x e_\lambda (z,s)ds$ and consider the $C^1$-functional ${\widetilde{\psi }}_\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\widetilde{\psi }}_\lambda (u)= \frac{1}{p}\vartheta (u) + \frac{\eta }{p}\Vert u\Vert _p^p - \int _\varOmega E_\lambda (z,u)dz \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

From (29) and since $\eta > \Vert \xi \Vert _\infty$, we see that ${\widetilde{\psi }}_\lambda$ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find $u_\lambda \in W^{1,p}(\varOmega )$ s.t.

$$\begin{aligned} {\widetilde{\psi }}_\lambda (u_\lambda )=\inf \left[ {\widetilde{\psi }}_\lambda (u) : u \in W^{1,p}(\varOmega ) \right] . \end{aligned}$$

(30)

Let $m_\mu =\min _{{\overline{\varOmega }}} {\overline{u}}_\mu >0$ (recall that ${\overline{u}}_\mu \in D_+$) and choose $t \in (0,1)$ small s.t. $t {\widehat{u}}_1(z) \le \min \{m_\mu ,\delta \}$ for all $z \in {\overline{\varOmega }}$ (here $\delta >0$ is as in hypothesis $H_1$(iii)). Because $q<p$ and by choosing $t \in (0,1)$ even smaller if necessary, we have that

$$\begin{aligned}&{\widetilde{\psi }}_\lambda (t {\widehat{u}}_1)<0,\\&\quad \Rightarrow {\widetilde{\psi }}_\lambda (u_\lambda )<0={\widetilde{\psi }}_\lambda (0) \quad \text{(see } (30))\\&\quad \Rightarrow u_\lambda \ne 0. \end{aligned}$$

From (30) we have

$$\begin{aligned}&{\widetilde{\psi }}'_\lambda (u_\lambda )=0, \nonumber \\&\quad \Rightarrow \langle A(u_\lambda ),h \rangle + \int _\varOmega (\xi (z)+\eta ) |u_\lambda |^{p-2}u_\lambda h dz + \int _{\partial \varOmega } \beta (z)|u_\lambda |^{p-2}u_\lambda h d \sigma \nonumber \\&\quad = \int _\varOmega e_\lambda (z,u_\lambda ) h dz\quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ). \end{aligned}$$

(31)

As in the proof of Proposition 9, using this time (31) and (29), we show that

$$\begin{aligned}&u_\lambda \in [0, {\overline{u}}_\mu ]=\{v \in W^{1,p}(\varOmega ): 0 \le v(z) \le {\overline{u}}_\mu (z) \text{ for } \text{ a.a. } z \in \varOmega \}, \quad u_\lambda \ne 0, \\&\quad \Rightarrow u_\lambda \in S(\lambda ) \subseteq D_+ \quad \text{(see } (29), (31)),\\&\quad \Rightarrow {\overline{u}}_\lambda \le {\overline{u}}_\mu . \end{aligned}$$

This proves that $\lambda \rightarrow {\overline{u}}_\lambda$ is nondecreasing.

Next we show the left continuity of this map. So, let $\{\lambda _n, \lambda \}_{n \ge 1} \subseteq {\mathcal {L}}$ and suppose that $\lambda _n \rightarrow \lambda ^-$. From the first part of the proof we have ${\overline{u}}_{\lambda _n} \le {\overline{u}}_\lambda$ for all $n \in {\mathbb {N}}$ and so we infer that $\{{\overline{u}}_{\lambda _n}\}_{n \ge 1} \subseteq W^{1,p}(\varOmega )$ is bounded. So, we may assume that

$$\begin{aligned} {\overline{u}}_{\lambda _n} \xrightarrow {w} {\widetilde{u}} \text{ in } W^{1,p}(\varOmega ) \quad \text{ and } \quad {\overline{u}}_{\lambda _n} \rightarrow {\widetilde{u}} \text{ in } L^{p}(\varOmega ) \text{ and } \text{ in } L^{p}(\partial \varOmega ). \end{aligned}$$

(32)

We have

$$\begin{aligned} \langle A({\overline{u}}_{\lambda _n}),h \rangle +\int _\varOmega \xi (z){\overline{u}}_{\lambda _n}^{p-1}h dz + \int _{\partial \varOmega } \beta (z){\overline{u}}_{\lambda _n}^{p-1}h d \sigma = \int _\varOmega (\lambda _n {\overline{u}}_{\lambda _n}^{p-1}+f(z,{\overline{u}}_{\lambda _n}))h dz\nonumber \\ \end{aligned}$$

(33)

for all $h \in W^{1,p}(\varOmega )$, all $n \in {\mathbb {N}}$. In (33) we choose $h= {\overline{u}}_{\lambda _n}-{\widetilde{u}} \in W^{1,p}(\varOmega )$, pass to the limit as $n \rightarrow +\infty$ and use (32). Then

$$\begin{aligned}&\lim _{n \rightarrow +\infty }\langle A({\overline{u}}_{\lambda _n}),{\overline{u}}_{\lambda _n} - {\widetilde{u}} \rangle =0,\nonumber \\&\quad \Rightarrow {\overline{u}}_{\lambda _n} \rightarrow {\widetilde{u}} \quad \text{ in } W^{1,p}(\varOmega ) \quad \text{(see } \text{ Proposition }~2). \end{aligned}$$

(34)

So, if in (33) we pass to the limit as $n \rightarrow +\infty$ and use (34), then

$$\begin{aligned} \langle A({\widetilde{u}}),h \rangle +\int _\varOmega \xi (z){\widetilde{u}}^{p-1}h dz + \int _{\partial \varOmega } \beta (z){\widetilde{u}}^{p-1}h d \sigma = \int _\varOmega (\lambda {\widetilde{u}}^{p-1}+f(z,{\widetilde{u}}))h dz\nonumber \\ \end{aligned}$$

(35)

for all $h \in W^{1,p}(\varOmega )$.

Set $B=\{\lambda _n\}_{n \ge 1}$ and let ${\widehat{c}}_7 \ge c_7({\widetilde{\lambda }})$ for all ${\widetilde{\lambda }} \in B$ (recall that $\lambda \rightarrow c_7(\lambda )$ is bounded on bounded sets). Consider ${\widehat{u}} \in D_+$ the unique positive solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varDelta _p u(z)+ \xi (z)u(z)^{p-1}=c_1 u(z)^{q-1}-{\widehat{c}}_7 u(z)^{r-1} &{}\quad \text{ in } \varOmega ,\\ \dfrac{\partial u}{\partial n_p} +\beta (z)u^{p-1}=0 &{}\quad \text{ on } \partial \varOmega , \end{array}\right. } \end{aligned}$$

(see Proposition 8 and the Remark following it).

We know that

$$\begin{aligned}&{\widehat{u}} \le {\overline{u}}_{\lambda _n} \quad \text{ for } \text{ all } n \in {\mathbb {N}},\\&\quad \Rightarrow {\widehat{u}} \le {\widetilde{u}}, \text{ that } \text{ is, } {\widetilde{u}} \ne 0. \end{aligned}$$

Then from (35) we infer that ${\widetilde{u}} \in S(\lambda )$.

Suppose that ${\widetilde{u}} \ne {\overline{u}}_\lambda$. Then we can find $z_0 \in {\overline{\varOmega }}$ s.t.

$$\begin{aligned} {\overline{u}}_\lambda (z_0) < {\widetilde{u}}(z_0). \end{aligned}$$

(36)

From Theorem 2 of Lieberman [13], we know that there exist $M>0$ and $\tau \in (0,1)$ s.t.

$$\begin{aligned} {\overline{u}}_{\lambda _n} \in C^{1,\tau }({\overline{\varOmega }})\quad \text{ and } \quad \Vert {\overline{u}}_{\lambda _n}\Vert _{C^{1,\tau }({\overline{\varOmega }})} \le M \quad \text{ for } \text{ all } n \in {\mathbb {N}}. \end{aligned}$$

(37)

Exploiting the compact embedding of $C^{1,\tau }({\overline{\varOmega }})$ into $C^{1}({\overline{\varOmega }})$ and using (34), from (37) we have

$$\begin{aligned}&{\overline{u}}_{\lambda _n} \rightarrow {\widetilde{u}}\quad \text{ in } C^{1}({\overline{\varOmega }}),\\&\quad \Rightarrow {\overline{u}}_{\lambda _n}(z_0) > {\overline{u}}_{\lambda }(z_0), \quad \text{ for } \text{ all } n \ge n_0 \quad \text{(see } (36)),\nonumber \end{aligned}$$

(38)

which contradicts the monotonicity of $\lambda \rightarrow {\overline{u}}_{\lambda }$ (recall $\lambda _n < \lambda$ for all $n \in {\mathbb {N}}$). Therefore ${\widetilde{u}}={\overline{u}}_{\lambda }$ and so from (38) we conclude that the map $\lambda \rightarrow {\overline{u}}_{\lambda }$ is left continuous from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$. $\square$

If we strengthen the conditions on the perturbation $f(z,\cdot )$, we can have uniqueness of the positive solution for problem $(P_{\lambda })$, $\lambda < {\widehat{\lambda }}_1$.

The new hypotheses on f(z, x) are the following:

$H_2$: $f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function s.t. for a.a. $z \in \varOmega$, $f(z,0)=0$, $f(z,x)>0$ for all $x >0$, hypotheses $H_2$(i), (ii), (iii) are the same as the corresponding hypotheses $H_1$(i), (ii), (iii) and

• (iv) for a.a. $z \in \varOmega$ the function $x \rightarrow \dfrac{f(z,x)}{x^{p-1}}$ is strictly decreasing on $(0,+\infty )$.

Example 2

The function $f_1(x)=x^{q-1}$ for all $x \ge 0$ with $1<q<p$ satisfies hypotheses $H_2$. On the other hand the function

$$\begin{aligned} f_2(x)={\left\{ \begin{array}{ll}x^{q-1}-x^{\tau -1} &{}\quad \text{ if } x \in [0,1],\\ \ln x^{p-1} &{}\quad \text{ if } 1<x,\end{array}\right. } \text{ with } 1<q<p, q< \tau , \end{aligned}$$

need not satisfy hypotheses $H_2$ unless additional restrictions are imposed on the exponents $q, \tau$.

Proposition 12

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_2$ hold and $\lambda < {\widehat{\lambda }}_1$, then problem $(P_{\lambda })$ has a unique positive solution $u_\lambda \in D_+$.

Proof

Existence follows from Proposition 7. The uniqueness is proved as in the proof of Proposition 8 using the nonlinear Picone’s identity (for an alternative approach, see the Remark following the proof of Proposition 8). $\square$

In this case, because of the uniqueness of the positive solution, Proposition 11 takes the following form:

Proposition 13

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_2$ hold, then the map $\lambda \rightarrow u_\lambda$ is nondecreasing and continuous from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$.

In fact, by strengthening hypothesis $H(\beta )_1$ (since we will use Proposition 4) and with an additional condition on the perturbation $f(z,\cdot )$ we can improve the monotonicity property of the maps $\lambda \rightarrow {\overline{u}}_\lambda$ in Proposition 11 and of the map $\lambda \rightarrow u_\lambda$ in Proposition 12.

So, we introduce the following conditions on the functions $\beta (z)$ and f(z, x):

$H(\beta )_2$: $\beta \in C^{0,\alpha }(\partial \varOmega )$ with $\alpha \in (0,1)$ and $\beta (z)>0$ for all $z \in \partial \varOmega$.

$H_3$: $f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function s.t. for a.a. $z \in \varOmega$, $f(z,0)=0$, $f(z,x)>0$ for all $x >0$, hypotheses $H_3$(i), (ii), (iii) are the same as the corresponding hypotheses $H_1$(i), (ii), (iii) and

• (iv) for every $\rho >0$, there exists ${\widehat{\xi }}_\rho >0$ s.t. for a.a. $z \in \varOmega$ the function $x \rightarrow f(z,x)+ {\widehat{\xi }}_\rho x^{p-1}$ is nondecreasing on $[0,\rho ]$.

We also introduce a corresponding strengthening of hypotheses $H_2$.

$H_4$: $f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function s.t. for a.a. $z \in \varOmega$, $f(z,0)=0$, $f(z,x)>0$ for all $x >0$, hypotheses $H_4$(i), (ii), (iii), (iv) are the same as the corresponding hypotheses $H_2$(i), (ii), (iii), (iv) and

• (v) for every $\rho >0$, there exists ${\widehat{\xi }}_\rho >0$ s.t. for a.a. $z \in \varOmega$, the function $x \rightarrow f(z,x)+ {\widehat{\xi }}_\rho x^{p-1}$ is nondecreasing on $[0,\rho ]$.

Proposition 14

If hypotheses $H(\xi )$, $H(\beta )_2$, $H_3$ hold, then the map $\lambda \rightarrow {\overline{u}}_\lambda$ is strictly increasing from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ in the sense that $\lambda < \mu \, \Rightarrow \, {\overline{u}}_\mu - {\overline{u}}_\lambda \in {\mathrm{int }}\, {\widehat{C}}_+$ with $D_0=\{z \in \partial \varOmega : {\overline{u}}_\mu (z)= {\overline{u}}_\lambda (z)\}$.

Proof

Let $\lambda ,\mu \in {\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ with $\lambda < \mu$. From Proposition 11 we know that

$$\begin{aligned} {\overline{u}}_\lambda \le {\overline{u}}_\mu . \end{aligned}$$

Let $\rho =\Vert {\overline{u}}_\mu \Vert _\infty$ and let ${\widehat{\xi }}_\rho >0$ be as postulated by hypothesis $H_3$(iv). We set

$$\begin{aligned} {\widetilde{\xi }}_\rho ={\widehat{\xi }}_\rho + \max \{-\mu ,0\}. \end{aligned}$$

For $\delta >0$ we define ${\overline{u}}_\lambda ^\delta = {\overline{u}}_\lambda + \delta \in D_+$. We have

$$\begin{aligned}&-\varDelta _p {\overline{u}}_\lambda ^\delta + (\xi (z)+{\widetilde{\xi }}_\rho )({\overline{u}}_\lambda ^\delta )^{p-1}\nonumber \\&\quad \le -\varDelta _p {\overline{u}}_\lambda + (\xi (z)+{\widetilde{\xi }}_\rho ){\overline{u}}_\lambda ^{p-1}+ \chi (\delta ) \quad \text{ with } \chi (\delta ) \rightarrow 0^+ \text{ as } \delta \rightarrow 0^+\nonumber \\&\quad = \lambda {\overline{u}}_\lambda ^{p-1}+f(z,{\overline{u}}_\lambda )+{\widetilde{\xi }}_\rho {\overline{u}}_\lambda ^{p-1}+ \chi (\delta )\nonumber \\&\quad = \mu {\overline{u}}_\lambda ^{p-1}+f(z,{\overline{u}}_\lambda )+{\widetilde{\xi }}_\rho {\overline{u}}_\lambda ^{p-1}-(\mu -\lambda ){\overline{u}}_\lambda ^{p-1} + \chi (\delta ). \end{aligned}$$

(39)

Note that if $\mu <0$, then ${\widetilde{\xi }}_\rho ={\widehat{\xi }}_\rho +|\mu |$ and we have

$$\begin{aligned}&0 \le \left[ f(z,{\overline{u}}_\mu ) + {\widehat{\xi }}_\rho {\overline{u}}_\mu ^{p-1}-(f(z,{\overline{u}}_\lambda )+{\widehat{\xi }}_\rho {\overline{u}}_\lambda ^{p-1})\right] +(|\mu |+\mu ) ({\overline{u}}_\mu ^{p-1}-{\overline{u}}_\lambda ^{p-1})\\&\quad \Leftrightarrow \mu {\overline{u}}_\lambda ^{p-1}+f(z,{\overline{u}}_\lambda )+{\widetilde{\xi }}_\rho {\overline{u}}_\lambda ^{p-1} \le \mu {\overline{u}}_\mu ^{p-1}+f(z,{\overline{u}}_\mu )+{\widetilde{\xi }}_\rho {\overline{u}}_\mu ^{p-1}. \end{aligned}$$

If $\mu \ge 0$, then ${\widetilde{\xi }}_\rho ={\widehat{\xi }}_\rho$ and using hypothesis $H_3$(iv) we have

$$\begin{aligned} \mu {\overline{u}}_\lambda ^{p-1} +f(z,{\overline{u}}_\lambda )+{\widehat{\xi }}_\rho {\overline{u}}_\lambda ^{p-1} \le \mu {\overline{u}}_\mu ^{p-1}+f(z,{\overline{u}}_\mu )+{\widehat{\xi }}_\rho {\overline{u}}_\mu ^{p-1}. \end{aligned}$$

Returning to (39), we have

$$\begin{aligned}&-\varDelta _p {\overline{u}}_\lambda ^\delta + (\xi (z)+{\widetilde{\xi }}_\rho )({\overline{u}}_\lambda ^\delta )^{p-1}\nonumber \\&\quad \le \mu {\overline{u}}_\mu ^{p-1}+f(z,{\overline{u}}_\mu )+{\widetilde{\xi }}_\rho {\overline{u}}_\mu ^{p-1}-(\mu -\lambda ){\overline{u}}_\lambda ^{p-1} + \chi (\delta )\nonumber \\&\quad = -\varDelta _p {\overline{u}}_\mu +{\widetilde{\xi }}_\rho {\overline{u}}_\mu ^{p-1}-(\mu -\lambda ){\overline{u}}_\lambda ^{p-1} + \chi (\delta ). \end{aligned}$$

(40)

Since $\mu >\lambda$ and ${\overline{u}}_\lambda \in D_+$, we have

$$\begin{aligned} 0 < {\widehat{m}} \le (\mu -\lambda ){\overline{u}}_\lambda (z)^{p-1}\quad \text{ for } \text{ all } z \in {\overline{\varOmega }}. \end{aligned}$$

Then since $\chi (\delta ) \rightarrow 0^+$ as $\delta \rightarrow 0^+$, for $\delta >0$ small we have

$$\begin{aligned} {\widehat{m}}-\chi (\delta )>0. \end{aligned}$$

Using this in (40) we have

$$\begin{aligned}&-\varDelta _p {\overline{u}}_\lambda ^\delta + (\xi (z)+{\widetilde{\xi }}_\rho )({\overline{u}}_\lambda ^\delta )^{p-1}< -\varDelta _p {\overline{u}}_\mu +(\xi (z) +{\widetilde{\xi }}_\rho ){\overline{u}}_\mu \\&\quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } \delta >0 \text{ small, }\\&\quad \Rightarrow {\overline{u}}_\mu - {\overline{u}}_\lambda \in {\mathrm{int }}\, {\widehat{C}}_+, \quad \text{(see } \text{ Proposition }~4 \text{ and } \text{ the } \text{ Remark } \text{ that } \text{ follows) }. \end{aligned}$$

In this case in the definition of ${\widehat{C}}_+$, $D_0=\{z \in \partial \varOmega : {\overline{u}}_\mu (z)={\overline{u}}_\lambda (z)\}$. $\square$

Similarly we have:

Proposition 15

If hypotheses $H(\xi )$, $H(\beta )_2$, $H_4$ hold, then the map $\lambda \rightarrow {\overline{u}}_\lambda$ is strictly increasing from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$.

The next theorem summarizes the situation for problem $(P_{\lambda })$ when the perturbation $f(z,\cdot )$ is $(p-1)$-sublinear.

Theorem 1

We have:

• 1. If hypotheses $H(\xi )$, $H(\beta )_1$, $H_1$ hold, then

  • (a) for all $\lambda \ge {\widehat{\lambda }}_1$ problem $(P_{\lambda })$ has no positive solution;

  • (b) for all $\lambda < {\widehat{\lambda }}_1$ problem $(P_{\lambda })$ has at least one positive solution and it admits a smallest positive solution ${\overline{u}}_\lambda \in D_+$;

  • (c) the map $\lambda \rightarrow {\overline{u}}_\lambda$ from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ is nondecreasing (that is, if $\lambda \le \mu$, then ${\overline{u}}_\lambda \le {\overline{u}}_\mu$) and left continuous.

• 2. If hypotheses $H(\xi )$, $H(\beta )_2$, $H_3$ hold, then the map $\lambda \rightarrow {\overline{u}}_\lambda$ from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ is strictly increasing as in Proposition 14.

• 3. If hypotheses $H(\xi )$, $H(\beta )_1$, $H_2$ hold and $\lambda < {\widehat{\lambda }}_1$, then problem $(P_{\lambda })$ has a unique solution $u_\lambda \in D_+$ and the map $\lambda \rightarrow u_\lambda$ from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ is nondecreasing and continuous.

• 4. If hypotheses $H(\xi )$, $H(\beta )_2$, $H_4$ hold, then the solution map $\lambda \rightarrow {\overline{u}}_\lambda$ from ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ is strictly increasing.

4 $(p-1)$-superlinear perturbation

In this section we consider the case where the perturbation $f(z,\cdot )$ is $(p-1)$-superlinear. In this case uniqueness of the solution fails and the problem exhibits a bifurcation-type behaviour, namely there are no positive solutions for all $\lambda \ge {\widehat{\lambda }}_1$ and there are at least two positive solutions for $\lambda < {\widehat{\lambda }}_1$.

The new hypotheses on the perturbation term f(z, x) are the following:

$H_5$: $f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function s.t. for a.a. $z \in \varOmega$ $f(z,0)=0$, $f(z,x)\ge 0$ for all $x > 0$, there exist $\varOmega _0 \subseteq \varOmega$ with $|\varOmega _0|_N>0$ s.t. $f(z,x)>0$ for all $z \in \varOmega _0$, all $x>0$ and

• (i) $f(z,x) \le a(z)(1+x^{r-1})$ for a.a. $z \in \varOmega$, all $x \ge 0$, with $a \in L^\infty (\varOmega )_+$, $r \in (p,p^*)$;

• (ii) if $F(z,x)=\int _0^xf(z,s)ds$, then

  $$\begin{aligned} \lim _{x \rightarrow + \infty }\dfrac{F(z,x)}{x^{p}}=+\infty \quad \text{ uniformly } \text{ for } \text{ a.a. } z\in \varOmega \end{aligned}$$

  and there exists $\tau \in (\max \{1,(r-p)\frac{N}{p}\},p^*)$ s.t.

  $$\begin{aligned} 0< {\widetilde{\xi }} \le \liminf _{x \rightarrow +\infty } \dfrac{f(z,x)x-pF(z,x)}{x^\tau } \quad \text{ uniformly } \text{ for } \text{ a.a. } z \in \varOmega ; \end{aligned}$$

• (iii) $\lim _{x \rightarrow 0^+}\dfrac{f(z,x)}{x^{p-1}}=0$ uniformly for a.a. $z \in \varOmega$.

Remark 4

As we did for the “sublinear” case, since we are looking for positive solutions and the above hypotheses concern the positive semiaxis, without any loss of generality, we may assume that $f(z,x)=0$ for a.a. $z \in \varOmega$, all $x < 0$. Hypothesis $H_5$(ii) implies that for a.a. $z \in \varOmega$ $f(z,\cdot )$ is $(p-1)$-superlinear. However, note that we do not use the usual in such cases “Ambrosetti–Rabinowitz condition” (the AR-condition for short, unilateral version since we are looking for positive solutions), which says that there exist $q>p$ and $M>0$ s.t.

$$\begin{aligned}&0<qF(z,x) \le f(z,x)x \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \ge M, \end{aligned}$$

(41)

$$\begin{aligned}&0< {\mathrm{ess}}\inf _{\varOmega } F(\cdot ,M) \end{aligned}$$

(42)

(see Ambrosetti–Rabinowitz [3] and Mugnai [15]). Integrating (41) and using (42) we obtain

$$\begin{aligned} c_{13}x^q \le F(z,x) \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \ge M, \text{ some } c_{13}>0. \end{aligned}$$

(43)

Hence from (41) and (43) we infer that near $+\infty$, $f(z,\cdot )$ exhibits at least $(q-1)$-polynomial growth. Our hypothesis $H_5$(ii) is more general. Indeed, suppose that the AR-condition holds. We may assume that $q > \max \{1, (r-p) \frac{N}{p}\}$. We have

$$\begin{aligned}&\frac{f(z,x)x-pF(z,x)}{x^q}=\frac{f(z,x)x-qF(z,x)}{x^q}+(q-p)\frac{F(z,x)}{x^q}\\&\quad \ge (q-p)\frac{F(z,x)}{x^q} \quad \text{(see } (41))\\&\quad \ge (q-p)c_{13}>0\quad \text{(see } (43)),\\&\quad \Rightarrow \liminf _{x \rightarrow + \infty } \frac{f(z,x)x-pF(z,x)}{x^q} \ge (q-p)c_{13}>0 \quad \text{ uniformly } \text{ for } \text{ a.a. } z \in \varOmega ,\\&\quad \Rightarrow \text{ hypothesis } H_5\text{(ii) } \text{ holds }. \end{aligned}$$

The function

$$\begin{aligned} f(x)=x^{p-1} \ln (1+x) \quad \text{ for } \text{ all } x\ge 0 \end{aligned}$$

satisfies hypotheses $H_5$, but not the AR-condition (see (41)).

From Propositions 5 and 6 we have

$$\begin{aligned} S(\lambda )&\subseteq D_+ \quad \text{ for } \text{ all } \lambda \in {\mathbb {R}},\\ S(\lambda )&= \emptyset \quad \text{ for } \text{ all } \lambda \ge {\widehat{\lambda }}_1. \end{aligned}$$

It follows that ${\mathcal {L}} \subseteq (-\infty , {\widehat{\lambda }}_1)$. In the next proposition we show that equality holds.

Proposition 16

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_5$ hold, then ${\mathcal {L}} = (-\infty , {\widehat{\lambda }}_1)$.

Proof

We fix $\lambda \in (-\infty , {\widehat{\lambda }}_1)$ and consider the Carathéodory function $k_\lambda (z,x)$ defined by

$$\begin{aligned} k_\lambda (z,x)= {\left\{ \begin{array}{ll} 0 &{} \quad \text{ if } x \le 0,\\ \lambda x^{p-1}+f(z,x) &{} \quad \text{ if } 0<x.\end{array}\right. } \end{aligned}$$

(44)

We set $K_\lambda (z,x)= \int _0^x k_\lambda (z,x)ds$ and consider the $C^1$-functional $w_\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} w_\lambda (u)= \frac{1}{p}\vartheta (u) + \frac{\eta }{p}\Vert u^-\Vert _p^p - \int _\varOmega K_\lambda (z,u)dz \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

As before $\eta > \Vert \xi \Vert _\infty$. Hypotheses $H_5$(i), (iii) imply that given $\varepsilon >0$, we can find $c_{14} =c_{14}(\varepsilon )>0$ s.t.

$$\begin{aligned} F(z,x) \le \frac{\varepsilon }{p}x^p+c_{14}x^r \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \ge 0. \end{aligned}$$

(45)

Choosing $\varepsilon \in (0, {\widehat{\lambda }}_1-\lambda )$ (recall $\lambda < {\widehat{\lambda }}_1$), for every $u \in W^{1,p}(\varOmega )$ we have

$$\begin{aligned} w_\lambda (u)&\ge \frac{1}{p}\left[ \vartheta (u^-)+\eta \Vert u^-\Vert _p^p\right] + \frac{1}{p}\vartheta (u^+) - \frac{\lambda +\varepsilon }{p}\Vert u^+\Vert _p^p-c_{14}\Vert u^+\Vert _r^r\nonumber \\&\quad \text{(see } (44) \text{ and } (45)).\nonumber \\&\ge c_{15} \Vert u\Vert ^p-c_{16}\Vert u\Vert ^r \quad \text{ for } \text{ some } c_{15},c_{16}>0,\nonumber \\&\quad \text{(use } \text{ Lemma }~1 \text{ and } \text{ recall } \eta > \Vert \xi \Vert _\infty ). \end{aligned}$$

(46)

Since $p<r$, from (46) we infer that $u=0$ is a strict local minimizer of $w_\lambda$. So, we can find $\rho \in (0,1)$ small s.t.

$$\begin{aligned} w_\lambda (0)=0 < \inf \left[ w_\lambda : \Vert u\Vert =\rho \right] =m^\lambda _\rho \end{aligned}$$

(47)

(see Aizicovici–Papageorgiou–Staicu [1], proof of Proposition 29).

Hypothesis $H_5$(ii) implies that

$$\begin{aligned} w_\lambda (t {\widehat{u}}_1) \rightarrow -\infty \quad \text{ as } t \rightarrow +\infty . \end{aligned}$$

(48)

Claim: $w_\lambda$ satisfies the C-condition.

Let $\{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega )$ be a sequence s.t.

$$\begin{aligned}&|w_\lambda (u_n)| \le M_1 \quad \text{ for } \text{ some } M_1>0, \text{ all } n \in {\mathbb {N}}, \end{aligned}$$

(49)

$$\begin{aligned}&(1+\Vert u_n\Vert )w'_\lambda (u_n) \rightarrow 0 \quad \text{ in } W^{1,p}(\varOmega )^* \text{ as } n \rightarrow +\infty . \end{aligned}$$

(50)

From (50) we have

$$\begin{aligned}&\bigg |\langle A(u_n),h\rangle + \int _\varOmega \xi (z)|u_n|^{p-2}u_n h dz + \int _{\partial \varOmega }\beta (z) |u_n|^{p-2}u_n h d \sigma \nonumber \\&\quad -\eta \int _\varOmega (u_n^-)^{p-1}h d \sigma -\int _\varOmega k_\lambda (z,u_n)hdz\bigg | \nonumber \\&\quad \le \frac{\varepsilon _n\Vert h\Vert }{1+\Vert u_n\Vert }\quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ) \text{ with } \varepsilon _n \rightarrow 0^+. \end{aligned}$$

(51)

In (51) we choose $h=-u_n^- \in W^{1,p}(\varOmega )$. Using (44) we obtain

$$\begin{aligned}&\left| \vartheta (u_n^-)+\eta \Vert u_n^-\Vert _p^p \right| \le \varepsilon _n \quad \text{ for } \text{ all } n \in {\mathbb {N}},\nonumber \\&\quad \Rightarrow c_{17} \Vert u_n^-\Vert ^p \le \varepsilon _n \quad \text{ for } \text{ all } n \in {\mathbb {N}}, \text{ some } c_{17}>0~\text{(recall } \text{ that } \eta >\Vert \xi \Vert _\infty ),\nonumber \\&\quad \Rightarrow u_n^- \rightarrow 0 \text{ in } W^{1,p}(\varOmega ). \end{aligned}$$

(52)

From (49), (52) and (44) it follows that

$$\begin{aligned} \vartheta (u_n^+)-\int _\varOmega [\lambda (u_n^+)^p+pF(z,u_n^+)]dz\le M_2 \quad \text{ for } \text{ some } M_2>0, \text{ all } n \in {\mathbb {N}}. \end{aligned}$$

(53)

In (51) we choose $h=u_n^+ \in W^{1,p}(\varOmega )$. Then

$$\begin{aligned} - \vartheta (u_n^+)+\int _\varOmega [\lambda (u_n^+)^p+f(z,u_n^+)u_n^+]dz\le \varepsilon _n \quad \text{ for } \text{ all } n \in {\mathbb {N}}. \end{aligned}$$

(54)

Adding (53) and (54) we obtain

$$\begin{aligned} \int _\varOmega [f(z,u_n^+)u_n^+ -pF(z,u_n^+)]dz\le M_3 \quad \text{ for } \text{ some } M_3>0, \text{ all } n \in {\mathbb {N}}. \end{aligned}$$

(55)

Hypotheses $H_5$(i), (ii) imply that we can find ${\widetilde{\xi }}_0 \in (0, {\widetilde{\xi }})$ and $c_{18}>0$ s.t.

$$\begin{aligned} {\widetilde{\xi }}_0 x^\tau -c_{18} \le f(z,x)x-pF(z,x) \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \ge 0. \end{aligned}$$

(56)

Using (56) in (55), we infer that

$$\begin{aligned} \{u_n^+\}_{n\ge 1} \subseteq L^\tau (\varOmega ) \quad \text{ is } \text{ bounded. } \end{aligned}$$

(57)

First suppose that $N > p$. Clearly in hypothesis $H_5$(ii), we can always assume that $\tau<r<p^*$ (recall that $p^*=+\infty$ if $p \ge N$). Let $t \in (0,1)$ be such that

$$\begin{aligned} \frac{1}{r}=\frac{1-t}{\tau }+\frac{t}{p^*}. \end{aligned}$$

(58)

From the interpolation inequality (see, for example, Gasiński–Papageorgiou [8] (p. 905)), we have

$$\begin{aligned}&\Vert u_n^+\Vert _r \le \Vert u_n^+\Vert _\tau ^{1-t}\Vert u_n^+\Vert _{p^*}^t,\nonumber \\&\quad \Rightarrow \Vert u_n^+\Vert ^r_r \le M_4\Vert u_n^+\Vert ^{tr} \quad \text{ for } \text{ some } M_4>0, \text{ all } n \in {\mathbb {N}}\\&\qquad \text{(see } (57) \text{ and } \text{ use } \text{ the } \text{ Sobolev } \text{ embedding } \text{ theorem). }\nonumber \end{aligned}$$

(59)

In (51) we choose $h=u_n^+ \in W^{1,p}(\varOmega )$. Then

$$\begin{aligned}&\vartheta (u_n^+)-\int _\varOmega [\lambda (u_n^+)^p+f(z,u_n^+)u_n^+]dz\le \varepsilon _n \quad \text{ for } \text{ all } n \in {\mathbb {N}}~\text{(see } (44)),\nonumber \\&\quad \Rightarrow \vartheta (u_n^+) \le c_{19}(1+\Vert u_n^+\Vert ^r_r) \quad \text{ for } \text{ some } c_{19}>0, \text{ all } n \in {\mathbb {N}}\nonumber \\&\qquad \text{(see } \text{ hypothesis } H_5\text{(i) } \text{ and } \text{ recall } \text{ that } r>p),\nonumber \\&\quad \Rightarrow \vartheta (u_n^+) \le c_{20}(1+\Vert u_n^+\Vert ^{tr}) \quad \text{ for } \text{ some } c_{20}>0, \text{ all } n \in {\mathbb {N}}~\text{(see } (59)). \end{aligned}$$

(60)

From hypothesis $H_5$(i) we see that we can always take $r \in (p,p^*)$ close to $p^*$ and as $r \rightarrow (p^*)^-$, we have $\tau >p$. So, there is no loss of generality in assuming that $\tau >p$. Then from (60) and (57), we have

$$\begin{aligned}&\vartheta (u_n^+) + \eta \Vert u_n^+\Vert _p^p \le c_{21}(1+\Vert u_n^+\Vert ^{tr}) \quad \text{ for } \text{ some } c_{21}>0, \text{ all } n \in {\mathbb {N}}, \nonumber \\&\quad \Rightarrow \Vert u_n^+\Vert ^p \le c_{22}(1+\Vert u_n^+\Vert ^{tr}) \, \text{ for } \text{ some } c_{22}>0, \text{ all } n \in {\mathbb {N}} \text{(recall } \text{ that } \eta > \Vert \xi \Vert _\infty ). \end{aligned}$$

(61)

From hypothesis $H_5$(ii) and (58) we see that

$$\begin{aligned}&tr <p,\nonumber \\&\quad \Rightarrow \{u_n^+\}_{n \ge 1} \subseteq W^{1,p}(\varOmega ) \quad \text{ is } \text{ bounded } \text{(see } (61)),\nonumber \\&\quad \Rightarrow \{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega ) \quad \text{ is } \text{ bounded } \text{(see } (52)). \end{aligned}$$

(62)

If $N\le p$, then $p^*=+\infty$, while the Sobolev embedding theorem says that $W^{1,p}(\varOmega ) \hookrightarrow L^q(\varOmega )$ for all $q \in [1,+\infty )$. Let $q>r>\tau$ and choose $t \in (0,1)$ s.t.

$$\begin{aligned}&\frac{1}{r}=\frac{1-t}{\tau }+\frac{t}{q},\nonumber \\&\quad \Rightarrow tr=\frac{q(r-\tau )}{q-\tau }. \end{aligned}$$

(63)

Note that

$$\begin{aligned} \frac{q(r-\tau )}{q-\tau } \rightarrow r-\tau \quad \text{ as } q \rightarrow p^*=+\infty . \end{aligned}$$

(64)

Since by hypothesis $H_5$(ii) we have $r-\tau <p$ (recall $N \le p$), for the previous argument (case $N \le p$) to work, we use $q>r$ big s.t. $tr<p$ (see (63), (64)). Then again we conclude that (62) holds. Because of (62) we may assume that

$$\begin{aligned} u_n \xrightarrow {w} u \text{ in } W^{1,p}(\varOmega ) \quad \text{ and } \quad u_n \rightarrow u \text{ in } L^p(\varOmega ) \text{ and } \text{ in } L^p(\partial \varOmega ). \end{aligned}$$

(65)

In (51) we choose $h=u_n-u \in W^{1,p}(\varOmega )$, pass to the limit as $n \rightarrow +\infty$ and use (65). Then we have

$$\begin{aligned}&\lim _{n \rightarrow +\infty } \langle A(u_n),u_n-u \rangle =0,\\&\quad \Rightarrow u_n \rightarrow u \quad \text{ in } W^{1,p}(\varOmega ) \quad \text{(see } \text{ Proposition }~2),\\&\quad \Rightarrow w_\lambda \text{ satisfies } \text{ the } C\text{-condition. } \end{aligned}$$

This proves the Claim.

Then (47), (48) and the Claim permit the use of the mountain pass theorem (see, for example, Gasiński–Papageorgiou [8] (p. 648)). So, we can find $u_\lambda \in W^{1,p}(\varOmega )$ s.t.

$$\begin{aligned} u_\lambda \in K_{w_\lambda }=\{v \in W^{1,p}(\varOmega ):w'_\lambda (v)=0\} \quad \text{ and } \quad w_\lambda (0)=0<m_\rho ^\lambda \le w_\lambda (u_\lambda ).\nonumber \\ \end{aligned}$$

(66)

From (66) it follows that $u_\lambda \ne 0$ and $u_\lambda \in S(\lambda ) \subseteq D_+$ (see Proposition 5). Therefore ${\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)$. $\square$

In fact as we did in the “sublinear” case, we can produce the minimal positive solution for problem $(P_{\lambda })$, $\lambda < {\widehat{\lambda }}_1$.

Proposition 17

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_5$ hold and $\lambda \in {\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)$, then problem ($P_\lambda$) has a smallest positive solution ${\overline{u}}_\lambda \in D_+$.

Proof

We argue as in the proof of Proposition 10. Recall that $S(\lambda )$ is downward directed (see Filippakis–Papageorgiou [5]). Using Lemma 3.10 of Hu–Papageorgiou [9] (p. 178), we can find a decreasing sequence $\{u_n\}_{n \ge 1} \subseteq S(\lambda )$ s.t.

$$\begin{aligned} \inf S(\lambda )= \inf _{n \ge 1} u_n. \end{aligned}$$

We have

$$\begin{aligned}&\langle A(u_n), h\rangle + \int _\varOmega \xi (z)u_n^{p-1}hdz + \int _{\partial \varOmega }\beta (z)u_n^{p-1}hd\sigma = \int _\varOmega [\lambda u_n^{p-1}+f(z,u_n)]hdz\\&\quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ), \text{ all } n \in {\mathbb {N}}.\nonumber \end{aligned}$$

(67)

In (67) we choose $h= u_n \in W^{1,p}(\varOmega )$, we obtain

$$\begin{aligned} \vartheta (u_n)= \lambda \Vert u_n\Vert ^p_p + \int _\varOmega f(z,u_n)u_ndz\quad \text{ for } \text{ all } n \in {\mathbb {N}}. \end{aligned}$$

(68)

Recall that

$$\begin{aligned} 0 \le u_n \le u_1 \in D_+ \quad \text{ for } \text{ all } n \in {\mathbb {N}}. \end{aligned}$$

(69)

From (68) to (69) it follows that

$$\begin{aligned} \{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega ) \quad \text{ is } \text{ bounded } \text{(see } \text{ hypotheses } H(\xi ), H(\beta )_1). \end{aligned}$$

So, we may assume that

$$\begin{aligned} u_n \xrightarrow {w} {\overline{u}}_\lambda \text{ in } W^{1,p}(\varOmega ) \quad \text{ and } \quad u_n \rightarrow {\overline{u}}_\lambda \text{ in } L^p(\varOmega ) \text{ and } \text{ in } L^p (\partial \varOmega ). \end{aligned}$$

(70)

In (67) we choose $h=u_n -{\overline{u}}_\lambda \in W^{1,p}(\varOmega )$, pass to the limit as $n \rightarrow + \infty$ and use (70). Then

$$\begin{aligned}&\lim _{n \rightarrow +\infty } \langle A(u_n), u_n-{\overline{u}}_\lambda \rangle =0, \nonumber \\&\quad \Rightarrow u_n \rightarrow {\overline{u}}_\lambda \text{ in } W^{1,p}(\varOmega )\quad \text{(see } \text{ Proposition }~2). \end{aligned}$$

(71)

Passing to the limit as $n \rightarrow + \infty$ in (67) and using (71), we obtain

$$\begin{aligned}&\langle A({\overline{u}}_\lambda ), h\rangle + \int _\varOmega \xi (z){\overline{u}}_\lambda ^{p-1}hdz + \int _{\partial \varOmega }\beta (z){\overline{u}}_\lambda ^{p-1}hd\sigma \\&\quad = \int _\varOmega [\lambda {\overline{u}}_\lambda ^{p-1}+f(z,{\overline{u}}_\lambda )]hdz \quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ),\\&\quad \Rightarrow {\overline{u}}_\lambda \text{ is } \text{ a } \text{ nonnegative } \text{ solution } \text{ of } \text{ problem } (P_\lambda ). \end{aligned}$$

If we can show that ${\overline{u}}_\lambda \ne 0$, then ${\overline{u}}_\lambda \in S(\lambda ) \subseteq D_+$. Arguing by contradiction, suppose that ${\overline{u}}_\lambda = 0$. Then

$$\begin{aligned} \Vert u_n\Vert \rightarrow 0 \quad \text{(see } (71)). \end{aligned}$$

We set $y_n=\dfrac{u_n}{\Vert u_n\Vert }$, $n \in {\mathbb {N}}$. Then for all $n \in {\mathbb {N}}$ we have $\Vert y_n\Vert =1$, $y_n \ge 0$. So, we may assume that

$$\begin{aligned} y_n \xrightarrow {w} y \text{ in } W^{1,p}(\varOmega ) \quad \text{ and } \quad y_n \rightarrow y \text{ in } L^p(\varOmega ) \text{ and } \text{ in } L^p (\partial \varOmega ). \end{aligned}$$

(72)

From (67) we have

$$\begin{aligned}&\langle A(y_n), h\rangle + \int _\varOmega \xi (z)y_n^{p-1}hdz + \int _{\partial \varOmega }\beta (z)y_n^{p-1}hd\sigma \\&\quad = \int _\varOmega \left[ \lambda y_n^{p-1}+\frac{N_f(u_n)}{\Vert u_n\Vert ^{p-1}}\right] hdz \quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ), \text{ all } n\in {\mathbb {N}}. \nonumber \end{aligned}$$

(73)

Here $N_f(y)(\cdot )= f(\cdot , y(\cdot ))$ for all $y \in W^{1,p}(\varOmega )$. We set $\rho =\Vert u_1\Vert _\infty$. Hypotheses $H_5$(i), (iii) imply that

$$\begin{aligned}&0 \le f(z,x) \le c_{23} x^{p-1} \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \in [0,\rho ], \text{ some } c_{23} >0,\\&\quad \Rightarrow \left\{ \frac{N_f(u_n)}{\Vert u_n\Vert ^{p-1}}\right\} _{n \ge 1} \subseteq L^p(\varOmega ) \quad \text{ is } \text{ bounded }. \end{aligned}$$

Then by passing to a suitable subsequence if necessary and using hypothesis $H_5$(iii), we have

$$\begin{aligned} \frac{N_f(u_n)}{\Vert u_n\Vert ^{p-1}} \xrightarrow {w} 0 \quad \text{ in } L^p(\varOmega ) \end{aligned}$$

(74)

(see Aizicovici–Papageorgiou–Staicu [1], proof of Proposition 14).

In (73) we choose $h = y_n -y \in W^{1,p}(\varOmega )$, pass to the limit as $n \rightarrow + \infty$ and use (72) and (74). Then

$$\begin{aligned}&\lim _{n \rightarrow +\infty } \langle A(y_n), y_n-y \rangle =0, \nonumber \\&\quad \Rightarrow y_n \rightarrow y \text{ in } W^{1,p}(\varOmega )\quad \text{(see } \text{ Proposition }~2), \ \Vert y\Vert =1, \ y\ge 0. \end{aligned}$$

(75)

So, if in (73) we pass to the limit as $n \rightarrow + \infty$ and use (74) and (75), then

$$\begin{aligned}&\langle A(y), h\rangle + \int _\varOmega \xi (z)y^{p-1}hdz + \int _{\partial \varOmega }\beta (z)y^{p-1}hd\sigma =\lambda \int _\varOmega y^{p-1}hdz\\&\quad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ). \end{aligned}$$

Choosing $h=y \in W^{1,p}(\varOmega )$, we obtain

$$\begin{aligned} \vartheta (y)= \lambda \Vert y\Vert ^p_p< {\widehat{\lambda }}_1 \Vert y\Vert ^p_p\quad \text{(see } (75) \text{ and } \text{ recall } \lambda < {\widehat{\lambda }}_1), \end{aligned}$$

a contradiction to Proposition 1. Therefore

$$\begin{aligned}&{\overline{u}}_\lambda \ne 0,\\&\quad \Rightarrow {\overline{u}}_\lambda \in S(\lambda ) \quad \text{ and } \quad {\overline{u}}_\lambda = \inf S(\lambda ). \end{aligned}$$

$\square$

As in the “sublinear” case, we have:

Proposition 18

If hypotheses $H(\xi )$, $H(\beta )_1$, $H_5$ hold, then the map $\lambda \rightarrow {\overline{u}}_\lambda$ from ${\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ is nondecreasing and left continuous.

Again by strengthening the conditions on the functions $\beta (\cdot )$ and $f(z, \cdot )$ we can improve the monotonicity of the map $\lambda \rightarrow {\overline{u}}_\lambda$.

The new hypotheses on the perturbation f(z, x) are the following:

$H_6$: $f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$ is a Carathéodory function s.t. for a.a. $z \in \varOmega$, $f(z,0)=0$, $f(z,x) \ge 0$ for all $x > 0$, there exists $\varOmega _0 \subseteq \varOmega$ with $f(z,x)>0$ for all $z \in \varOmega _0$, all $x>0$, hypotheses $H_6$(i), (ii), (iii) are the same as the corresponding hypotheses $H_5$(i), (ii), (iii) and

• (iv) for every $\rho >0$, there exists ${\widehat{\xi }}_\rho >0$ s.t. for a.a. $z \in \varOmega$, the function

  $$\begin{aligned} x \rightarrow f(z,x)+ {\widehat{\xi }}_\rho x^{p-1} \end{aligned}$$

  is nondecreasing on $[0,\rho ]$.

Proposition 19

If hypotheses $H(\xi )$, $H(\beta )_2$, $H_6$ hold, then the map $\lambda \rightarrow {\overline{u}}_\lambda$ from ${\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ is strictly decreasing.

In fact under these stronger conditions on $\beta (z)$ and f(z, x), we can produce a second positive solution for problem $(P_{\lambda })$, when $\lambda \in {\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)$.

Proposition 20

If hypotheses $H(\xi )$, $H(\beta )_2$, $H_6$ hold and $\lambda \in {\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)$, then problem ($P_\lambda$) admits at least two positive solutions

$$\begin{aligned} u_\lambda , {\widehat{u}}_\lambda \in D_+, \quad u_\lambda \le {\widehat{u}}_\lambda , \quad u_\lambda \ne {\widehat{u}}_\lambda . \end{aligned}$$

Proof

From Proposition 16 we already have a positive solution $u_\lambda \in D_+$. We may assume that $u_\lambda$ is the minimal positive solution, that is, $u_\lambda = {\overline{u}}_\lambda$ (see Proposition 17). We introduce the following Carathéodory function

$$\begin{aligned} \zeta _\lambda (z,x)={\left\{ \begin{array}{ll}(\lambda +\eta )u_\lambda (z)^{p-1}+f(z,u_\lambda (z)) &{}\quad \text{ if } x \le u_\lambda (z),\\ (\lambda +\eta ) x^{p-1}+f(z,x) &{}\quad \text{ if } u_\lambda (z) <x,\end{array}\right. } \end{aligned}$$

(76)

with $\eta > \Vert \xi \Vert _\infty$ as always. We set $Z_\lambda (z,x)= \int _0^x \zeta _\lambda (z,s)ds$ and consider the $C^1$-functional $j_\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} j_\lambda (u)=\frac{1}{p}\vartheta (u) +\frac{\eta }{p}\Vert u\Vert ^p_p- \int _\varOmega Z_\lambda (z,u)dz, \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

From (76) it is clear that

$$\begin{aligned} j_\lambda = w_\lambda +\xi _\lambda ^*\quad \text{ with } \xi _\lambda ^*\in {\mathbb {R}} \end{aligned}$$

(77)

with $w_\lambda \in C^1(W^{1,p}(\varOmega ))$ as in the proof of Proposition 16. From (77) and the Claim in the proof of Proposition 16, it follows that

$$\begin{aligned} j_\lambda \text{ satisfies } \text{ the } C\text{-condition }. \end{aligned}$$

(78)

Claim: We may assume that $u_\lambda \in D_+$ is a local minimizer of $j_\lambda$.

Let $\lambda<\mu <{\widehat{\lambda }}_1$ and let $u_\mu \in S(\mu ) \subseteq D_+$ (see Proposition 16). We consider the following truncation of $\zeta _\lambda (z, \cdot )$

$$\begin{aligned} {\widehat{\zeta }}_\lambda (z,x)={\left\{ \begin{array}{ll}\zeta (z,x) &{}\quad \text{ if } x \le u_\mu (z),\\ \zeta (z,u_\mu (z)) &{} \quad \text{ if } u_\mu (z) <x.\end{array}\right. } \end{aligned}$$

(79)

Evidently this is a Carathéodory function. We set ${\widehat{Z}}_\lambda (z,x)= \int _0^x {\widehat{\zeta }}_\lambda (z,s)ds$ and consider the $C^1$-functional ${\widehat{j}}_\lambda : W^{1,p} (\varOmega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\widehat{j}}_\lambda (u)=\frac{1}{p}\vartheta (u) +\frac{\eta }{p}\Vert u\Vert ^p_p- \int _\varOmega {\widehat{Z}}_\lambda (z,u)dz \quad \text{ for } \text{ all } u \in W^{1,p}(\varOmega ). \end{aligned}$$

If $K_{{\widehat{j}}_\lambda }= \{u \in W^{1,p}(\varOmega ): {\widehat{j}}_\lambda ^\prime (u)=0\}$, then we will show that

$$\begin{aligned} K_{{\widehat{j}}_\lambda } \subseteq [u_\lambda , u_\mu ]= \{u \in W^{1,p}(\varOmega ): u_\lambda (z) \le u(z) \le u_\mu (z) \ \text{ for } \text{ a.a. } z \in \varOmega \}. \end{aligned}$$

So, let $u \in K_{{\widehat{j}}_\lambda }$. Then

$$\begin{aligned}&{\widehat{j}}_\lambda ^\prime (u)=0 \nonumber \\&\quad \Rightarrow \langle A(u), h\rangle + \int _\varOmega (\xi (z)+\eta )|u|^{p-2}uhdz + \int _{\partial \varOmega }\beta (z)|u|^{p-2}uhd\sigma = \int _\varOmega {\widehat{\zeta }}_\lambda (z,u)hdz\\&\qquad \text{ for } \text{ all } h \in W^{1,p}(\varOmega ).\nonumber \end{aligned}$$

(80)

In (80) we choose $h=(u_\lambda -u)^+ \in W^{1,p}(\varOmega )$. Then

$$\begin{aligned}&\langle A(u), (u_\lambda -u)^+\rangle + \int _\varOmega (\xi (z)+\eta )|u|^{p-2}u(u_\lambda -u)^+dz + \int _{\partial \varOmega }\beta (z)|u|^{p-2}u(u_\lambda -u)^+d\sigma \\&\quad = \int _\varOmega [(\lambda +\eta )u_\lambda ^{p-1}+f(z,u_\lambda )](u_\lambda -u)^+dz \quad \text{(see } (76) \text{ and } (79))\\&\quad = \langle A(u_\lambda ), (u_\lambda -u)^+\rangle + \int _\varOmega (\xi (z)+\eta )u_\lambda ^{p-1}(u_\lambda -u)^+dz \\&\qquad + \int _{\partial \varOmega }\beta (z)u_\lambda ^{p-1}(u_\lambda -u)^+d\sigma \quad \text{(since } u_\lambda \in S(\lambda )),\\&\quad \Rightarrow \langle A(u_\lambda )-A(u), (u_\lambda -u)^+\rangle + \int _\varOmega (\xi (z)+\eta )(u_\lambda ^{p-1}-|u|^{p-2}u)(u_\lambda -u)^+dz\\&\qquad + \int _{\partial \varOmega }\beta (z)(u_\lambda ^{p-1}-|u|^{p-2}u)(u_\lambda -u)^+d\sigma =0,\\&\quad \Rightarrow u_\lambda \le u \quad \text{(recall } \text{ that } \eta > \Vert \xi \Vert _\infty \text{ and } \text{ see } \text{ hypothesis } H(\beta )). \end{aligned}$$

Also in (80), we choose $h=(u-u_\mu )^+ \in W^{1,p}(\varOmega )$. Then

$$\begin{aligned}&\langle A(u), (u-u_\mu )^+\rangle + \int _\varOmega (\xi (z)+\eta )u^{p-1}(u-u_\mu )^+dz + \int _{\partial \varOmega }\beta (z)u^{p-1}(u-u_\mu )^+d\sigma \\&\quad = \int _\varOmega [(\lambda +\eta )u_\mu ^{p-1}+f(z,u_\mu )](u-u_\mu )^+dz \quad \text{(see } (76), (79))\\&\quad \le \int _\varOmega [(\mu +\eta )u_\mu ^{p-1}+f(z,u_\mu )](u-u_\mu )^+dz \quad \text{(since } \lambda <\mu )\\&\quad = \langle A(u_\mu ), (u -u_\mu )^+\rangle + \int _\varOmega (\xi (z)+\eta )u_\mu ^{p-1}(u-u_\mu )^+dz + \int _{\partial \varOmega }\beta (z)u_\mu ^{p-1}(u-u_\mu )^+d\sigma \\&\qquad \text{(since } u_\mu \in S(\mu )),\\&\quad \Rightarrow \langle A(u)-A(u_\mu ), (u-u_\mu )^+\rangle + \int _\varOmega (\xi (z)+\eta )(u^{p-1}-u_\mu ^{p-1})(u-u_\mu )^+dz\\&\qquad + \int _{\partial \varOmega }\beta (z)(u^{p-1}-u_\mu ^{p-1})(u-u_\mu )^+d\sigma \le 0,\\&\quad \Rightarrow u \le u_\mu . \end{aligned}$$

So, we have proved that

$$\begin{aligned}&u \in [u_\lambda ,u_\mu ],\nonumber \\&\quad \Rightarrow K_{{\widehat{j}}_\lambda } \subseteq [u_\lambda ,u_\mu ]. \end{aligned}$$

(81)

Since $\eta > \Vert \xi \Vert _\infty$, from (76) and (79) it follows that ${\widehat{j}}_\lambda$ is coercive. Also, the Sobolev embedding theorem and the compactness of the trace map imply that ${\widehat{j}}_\lambda$ is sequentially weakly lower semicontinuous. So, from the Weierstrass-Tonelli theorem, we can find ${\widetilde{u}}_\lambda \in W^{1,p}(\varOmega )$ s.t.

$$\begin{aligned}&{\widehat{j}}_\lambda ({\widetilde{u}}_\lambda ) = \inf \left[ {\widehat{j}}_\lambda (u) : u\in W^{1,p}(\varOmega ) \right] , \nonumber \\&\quad \Rightarrow {\widetilde{u}}_\lambda \in K_{{\widehat{j}}_\lambda } \subseteq [u_\lambda ,u_\mu ]\quad \text{(see } (81)). \end{aligned}$$

(82)

If ${\widetilde{u}}_\lambda \ne u_\lambda$, then from (76), (79) and (82), we see that

$$\begin{aligned} {\widetilde{u}}_\lambda \in S(\lambda ) \subseteq D_+, \quad u_\lambda \le {\widetilde{u}}_\lambda , \quad {\widetilde{u}}_\lambda \ne u_\lambda . \end{aligned}$$

So, this is the desired second solution of $(P_{\lambda })$ and we are done.

Therefore, we assume that ${\widetilde{u}}_\lambda = u_\lambda$. From Proposition 19, we have

$$\begin{aligned} u_\mu -u_\lambda \in {\mathrm{int }}\, {\widehat{C}}_+ \quad \text{(recall } \text{ that } u_\lambda = {\overline{u}}_\lambda ). \end{aligned}$$

(83)

From (76) and (79) it is clear that

$$\begin{aligned} {\widehat{j}}_\lambda \big |_{[0,u_\mu ]} = j_\lambda \big |_{[0,u_\mu ]}. \end{aligned}$$

From this equality and (83) we infer that

$$\begin{aligned}&u_\lambda \text{ is } \text{ a } \text{ local } C^1({\overline{\varOmega }})\text{-minimizer } \text{ of } j_\lambda ,\\&\quad \Rightarrow u_\lambda \text{ is } \text{ a } \text{ local } W^{1,p}(\varOmega )\text{-minimizer } \text{ of } j_\lambda \quad \text{(see } \text{ Proposition }~3). \end{aligned}$$

This proves the Claim.

In proving (81), we established that

$$\begin{aligned} K_{j_\lambda } \subseteq [u_\lambda ) = \{ u \in W^{1,p}(\varOmega ) : u_\lambda (z) \le u(z) \text{ for } \text{ a.a. } z \in \varOmega \}. \end{aligned}$$

(84)

We assume that $K_{j_\lambda }$ is finite or otherwise (84) implies that we already have a whole sequence of distinct positive solutions of $(P_{\lambda })$, all bigger that $u_\lambda$, hence we are done. Then we can find $\rho \in (0,1)$ small s.t.

$$\begin{aligned} j_\lambda (u_\lambda ) < \inf \left[ j_\lambda (u) : \Vert u-u_\lambda \Vert = \rho \right] =m^\lambda _\rho \end{aligned}$$

(85)

(see Aizicovici–Papageorgiou–Staicu [1], proof of Proposition 29). Note that hypothesis $H_6$(ii) and (76) imply that

$$\begin{aligned} j_\lambda (t {\widehat{u}}_1) \rightarrow - \infty \quad \text{ as } t \rightarrow + \infty . \end{aligned}$$

(86)

From (78), (85) and (86) we see that we can apply the mountain pass theorem and find ${\widehat{u}}_\lambda \in W^{1,p}(\varOmega )$ s.t.

$$\begin{aligned} {\widehat{u}}_\lambda \in K_{j_\lambda }\quad \text{ and } \quad m^\lambda _\rho \le j_\lambda ({\widehat{u}}_\lambda ). \end{aligned}$$

(87)

From (76), (84), (85) and (87), we infer that

$$\begin{aligned} {\widehat{u}}_\lambda \in S(\lambda ) \subseteq D_+, \quad u_\lambda \le {\widehat{u}}_\lambda , \quad {\widehat{u}}_\lambda \ne u_\lambda .\nonumber \\ \end{aligned}$$

$\square$

So, summarizing the situation for problem $(P_{\lambda })$ when the perturbation $f(z, \cdot )$ is $(p-1)$-superlinear, we have the following theorem

Theorem 2

If hypotheses $H(\xi )$, $H(\beta )_2$, $H_6$ hold, then

• (a) for all $\lambda \ge {\widehat{\lambda }}_1$ problem $(P_{\lambda })$ has no positive solution;

• (b) for all $\lambda < {\widehat{\lambda }}_1$ problem $(P_{\lambda })$ has at least two positive solutions

  $$\begin{aligned} u_\lambda , {\widehat{u}}_\lambda \in D_+, \quad u_\lambda \le {\widehat{u}}_\lambda , \quad {\widehat{u}}_\lambda \ne u_\lambda ; \end{aligned}$$

• (c) for all $\lambda < {\widehat{\lambda }}_1$ problem $(P_{\lambda })$ has a smallest positive solution ${\overline{u}}_\lambda \in D_+$ and the map $\lambda \rightarrow {\overline{u}}_\lambda$ from ${\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)$ into $C^1({\overline{\varOmega }})$ is strictly increasing and left continuous.