Abstract

We prove existence and multiplicity of positive solutions for semipositone problems involving 𝑝-Laplacian in a bounded smooth domain of ℝ𝑁 under the cases of sublinear and superlinear nonlinearities term.

1. Introduction

In this paper, we shall study the following semipositone problem involving the 𝑝-Laplacian: ||||−div∇𝑢𝑝−2∇𝑢=𝜆𝑓(𝑢)inΩ,𝑢=0on𝜕Ω,(1.1) where Ω⊂ℝ𝑁 is a smooth bounded domain, 𝜆 is a positive parameter, and 𝑓∶[0,+∞)→ℝ is a continuous function satisfying the condition(F0)𝑓(0)=−𝑎<0.

Such problems are usually referred in the literature as semipositone problems. We refer the reader to [1], where Castro and Shivaji initially called them nonpositone problems, in contrast with the terminology positone problems, coined by Keller and Cohen in [2], when the nonlinearity 𝑓 was positive and monotone.

Under the case of 𝑝≡2, a novel variational approach is presented by Costa et al. [3] to the question of existence and multiplicity of positive solutions to problem (1.1), where they consider both the sublinear and superlinear cases. The aim of this paper is to extend their results to the case of 𝑝-Laplacian. The main difficulty is in verifying (PS)𝑐-condition because of the operator −Δ𝑝 is not self-adjoint linear.

We define the discontinuous nonlinearity 𝑔(𝑠) by 𝑔(𝑠)=0,if𝑠≤0,𝑓(𝑠),if𝑠>0.(1.2) We shall consider the modified problem ||||−div∇𝑢𝑝−2∇𝑢=𝜆𝑔(𝑢)inΩ,𝑢=0on𝜕Ω.(1.3) We note that the set of positive solutions of (1.1) and (1.3) do coincide.

Our main results concerning problems (1.1) and (1.3) are the following:

Theorem 1.1. Assume (F0) and the following assumptions:(F1)lim𝑠→+∞(𝑓(𝑠)/𝑠𝑝−1)=0 (the sublinear case);(F2)𝐹(𝛿)>0 for some 𝛿>0, where ∫𝐹(𝑢)=𝑢0𝑓(𝑠)𝑑𝑠.Then, there exist 0<𝜆0≤𝜆∗ such that (1.3) has no nontrivial nonnegative solution for 0<𝜆<𝜆0, and has at least two nontrivial nonnegative solutions 𝑢𝜆, 𝑣𝜆 for all 𝜆>𝜆∗. Moreover, when Ω is a ball 𝐵𝑅=𝐵𝑅(0), these two solutions are non-increasing, radially symmetric and, if 𝑁≥2, at least one of them is positive, hence a solution of (1.1).

Theorem 1.2. Assume (F0), (F2) and the following assumptions:(F3)lim𝑠→+∞(𝑓(𝑠)/𝑠𝑝−1)=+∞, lim𝑠→+∞(𝑓(𝑠)/𝑠𝑝∗−2)=0 (the sublinear, subcritical case);(F4)𝜃𝐹(𝑠)≤𝑓(𝑠)𝑠for all 𝑠≥𝐾 and some 𝜃>𝑝 (the AR-condition).Then, (1.3) has at least one nonnegative solution 𝑣𝜆 for all 𝜆>0. IfΩ=𝐵𝑅 then 𝑣𝜆 is nonincreasing, radially symmetric and one of the two alternatives occurs. (i)There exists 𝜆∗>0 such that, for all 0<𝜆<𝜆∗, 𝑣𝜆 is a positive solution of (1.1) having negative normal derivative on 𝜕𝐵𝑅. (ii)For some sequence 𝜇𝑛→0, problem (1.1) with 𝜆≡𝜇𝑛 has a positive solution 𝑤𝜇𝑛 with zero normal derivative on 𝜕𝐵𝑅.

2. Preliminaries

We start by recalling some basic results on variational methods for locally Lipschitz functionals. Let (𝑋,‖⋅‖) be a real Banach space and 𝑋∗ is its topological dual. A function 𝑓∶𝑋→ℝ is called locally Lipschitz if each point 𝑢∈𝑋 possesses a neighborhood Ω𝑢 such that |𝑓(𝑢1)−𝑓(𝑢2)|≤𝐿‖𝑢1−𝑢2‖ for all 𝑢1,𝑢2∈Ω𝑢, for a constant 𝐿>0 depending on Ω𝑢. The generalized directional derivative of 𝑓 at the point 𝑢∈𝑋 in the direction 𝑣∈𝑋 is 𝑓0(𝑢,𝑣)=limsup𝑤→𝑢,𝑡→0+1𝑡(𝑓(𝑤+𝑡𝑣)−𝑓(𝑤)).(2.1) The generalized gradient of 𝑓 at 𝑢∈𝑋 is defined by 𝑢𝜕𝑓(𝑢)=∗∈𝑋∗∶⟨𝑢∗,𝜑⟩≤𝑓0,(𝑢;𝜑)∀𝜑∈𝑋(2.2) which is a nonempty, convex, and 𝑤∗-compact subset of 𝑋, where ⟨⋅,⋅⟩ is the duality pairing between 𝑋∗ and 𝑋. We say that 𝑢∈𝑋 is a critical point of 𝑓 if 0∈𝜕𝑓(𝑢). For further details, we refer the reader to Chang [4].

We list some fundamental properties of the generalized directional derivative and gradient that will be possibly used throughout the paper.

Proposition 2.1 (see [4, 5]). (1)  Let 𝑗∶𝑋→ℝ be a continuously differentiable function. Then 𝜕𝑗(𝑢)={𝑗′(𝑢)}, 𝑗0(𝑢;𝑧) coincides with ⟨𝑗′(𝑢),𝑧⟩𝑋, and (𝑓+𝑗)0(𝑢,𝑧)=𝑓0(𝑢;𝑧)+⟨𝑗′(𝑢),𝑧⟩𝑋for all 𝑢,𝑧∈𝑋.
(2)  𝜕(𝜆𝑓)(𝑢)=𝜆𝜕(𝑢) for all 𝜆∈ℝ.
(3)  If 𝑓 is a convex functional, then 𝜕𝑓(𝑢) coincides with the usual subdifferential of 𝑓 in the sense of convex analysis.
(4)  If 𝑓 has a local minimum (or a local maximum) at 𝑢0∈𝑋, then 0∈𝜕𝑓(𝑢0).
(5)  ‖𝜉‖𝑋∗≤𝐿  for all 𝜉∈𝜕𝑓(𝑢).
(6)  𝑓0(𝑢,ℎ)=max{⟨𝜉,ℎ⟩∶𝜉∈𝜕𝑓(𝑢)} for all ℎ∈𝑋.
(7)  The function 𝑚(𝑢)=min𝑤∈𝜕𝑓(𝑢)‖𝑤‖𝑋∗,(2.3) exists and is lower semicontinuous; that is, lim𝑢→𝑢0inf𝑚(𝑢)≥𝑚(𝑢0).

In the following we need the nonsmooth version of the Palais-Smale condition.

Definition 2.2. One says 𝜑 that nonsmooth satisfies the (PS)𝑐-condition if any sequence {𝑢𝑛}⊂𝑋 such that 𝜑(𝑢𝑛)→𝑐 and 𝑚(𝑢𝑛)→0.
In what follows we write the (PS)𝑐-condition as simply as the PS-condition if it holds for every level 𝑐∈ℝ for the Palais-Smale condition at level 𝑐.
We note that property (4) above says that a local minimum (or local maximum) of 𝜑 is a critical point of 𝜑.Finally, we point out that many of the results of the classical critical point theory have been extended by Chang [4] to this setting of locally Lipschitz functionals. For example, one has the following celebrated theorem.

Theorem 2.3 (nonsmooth mountain pass theorem; see [4, 5]). If 𝑋 is a reflexive Banach space, 𝜑∶𝑋→ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS)𝑐-condition, and for some 𝑟>0 and 𝑒1∈𝑋 with ‖𝑒1‖>𝑟, max{𝜑(0),𝜑(𝑒1)}≤inf{𝜑(𝑢)∶‖𝑢‖=𝑟}. Then 𝜑 has a nontrivial critical 𝑢∈𝑋 such that the critical value 𝑐=𝜑(𝑢) is characterized by the following minimax principle: 𝑐=inf𝛾∈Γmax[]𝑡∈0,1𝜑(𝛾(𝑡)),(2.4) where Γ={𝛾∈𝐶([0,1],𝑋)∶𝛾(0)=0,𝛾(1)=𝑒1}.

We would like to point out that we can obtain the same results for problem (1.3) through approximation of the discontinuous nonlinearity by a sequence of continuous functions. Variational methods were then applied to the corresponding sequence of problems and limits were taken. For the rest of this paper, we write 𝑋=𝑊01,𝑝(Ω) with the norm by ∫‖𝑢‖=(Ω|∇𝑢|𝑝𝑑𝑥)1/𝑝 and denote by 𝑐 and 𝑐𝑖 the generic positive constants for simplicity.

3. Proof of the Main Results

Now, having listed some basic results on critical point theory for the Lipschitz functionals, let us consider the functional 𝜑𝜆(1𝑢)=𝑝Ω||||∇𝑢𝑑𝑥−𝜆Ω𝐺(𝑢)𝑑𝑥,(3.1) where ∫𝐺(𝑢)=𝑢0𝑔(𝑠)𝑑𝑠 and 𝑔(𝑠) were defined in (1.2). Clearly 𝐺∶ℝ→ℝ is a locally Lipschitz continuous function and satisfies 𝐺(𝑠)=0 for 𝑠≤0. In view of [3, Theorems 2.1 and 2.2], the above formula for 𝜑𝜆(𝑢) defines a locally Lipschitz functional on 𝑋 whose critical points are solutions of the differential inclusion ||||−div∇𝑢𝑝−2∇𝑢∈𝜕𝐺(𝑢)a.e.inΩ.(3.2) In our present case, it follows that 𝜕𝐺(𝑠)=𝑓(𝑠) for 𝑠>0, 𝜕𝐺(𝑠)=0 for 𝑠<0, and 𝜕𝐺(𝑠)=[−𝑎,0] for 𝑠=0.

We start with some preliminary lemmas.

Lemma 3.1. Assume (F0), (F1), and (F2). Then there exists 𝜆0 such that problem (1.3) has no nontrivial solution 0≤𝑢∈𝑋 for 0<𝜆<𝜆0.

Proof . If 𝑢≥0 is a solution of problem (1.3), then, multiplying the equation by 𝑢 and integrating over Ω yields ‖𝑢‖𝑝=Ω||||∇𝑢𝑝𝑑𝑥=𝜆Ω𝑔(𝑢)𝑢𝑑𝑥=𝜆{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}𝑢𝑔(𝑢)𝑑𝑥+{𝑥∈Ω∣𝑢(𝑥)≥𝛿0},𝑢𝑔(𝑢)𝑑𝑥(3.3) hence ‖𝑢‖𝑝≤𝜆{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}𝑢𝑔(𝑢)𝑑𝑥,(3.4) where we have chosen 𝛿0>0 so that 𝑔(𝑠)≤0 for 0≤𝑠≤𝛿0 (such a 𝛿0 exists in view of (F0)). Now, since (F1) implies the existence of 𝑐>0 such that 𝑠𝑔(𝑠)≤𝑐(1+𝑠𝑝),(3.5) for all 𝑠>0, we obtain from (3.4) that ‖𝑢‖𝑝≤𝜆𝑐{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}(1+𝑢𝑝1)𝑑𝑥≤𝜆𝑐1+𝛿𝑝0{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}𝑢𝑝𝑑𝑥≤𝜆𝑐1Ω𝑢𝑝𝑑𝑥,(3.6) so that ‖𝑢‖𝑝≤𝜆𝑐2‖𝑢‖𝑝,(3.7) where this last constant 𝑐2>0 is independent of both 𝑢 and 𝜆. Therefore we must have 1𝜆≥𝑐2∶=𝜆0>0.(3.8)

Lemma 3.2. Assume (F0) and either (F1) or (F3). Then 𝑢=0 is a strict local minimum of the functional 𝜑𝜆.

Proof . Since (F1) or (F3) implies the existence of 𝑐3>0 such that 𝐺(𝑠)≤𝑐31+|𝑠|𝑝∗∀𝑠∈ℝ,(3.9) recall also that 𝐺(𝑠)=0 for 𝑠≤0. Then, with 𝛿0>0 as in the proof of Lemma 3.1 and noticing that 𝐺(𝑠)≤0 for all −∞<𝑠≤𝛿0, we can write for an arbitrary 𝑢∈𝑋, 𝜑𝜆(1𝑢)=𝑝‖𝑢‖𝑝−𝜆Ω≥1𝐺(𝑢)𝑑𝑥𝑝‖𝑢‖𝑝−𝜆{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}≥1𝐺(𝑢)𝑑𝑥𝑝‖𝑢‖𝑝−𝜆𝑐3{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}1+𝑢𝑝∗≥1𝑑𝑥𝑝‖𝑢‖𝑝−𝜆𝑐311+𝛿𝑝∗0{𝑥∈Ω∣𝑢(𝑥)≥𝛿0}𝑢𝑝∗𝑑𝑥,(3.10) so that, using the Sobolev embedding theorem in the last inequality and with a constant 𝑐4>0 independent of 𝑢 and Ω, we obtain 𝜑𝜆1(𝑢)≥𝑝‖𝑢‖𝑝−𝜆𝑐4‖𝑢‖𝑝∗=1𝑝‖𝑢‖𝑝1−𝑝𝜆𝑐4‖𝑢‖𝑝∗−𝑝.(3.11) Therefore, for each 0<𝜌<𝜌𝜆∶=1/(𝑝𝜆𝑐4)𝑝∗−𝑝, it follows that 𝜑𝜆(𝑢)≥𝛼𝜌>0 if ‖𝑢‖=𝜌. This shows that 𝑢=0 is a strict local minimum of 𝜑𝜆.

Lemma 3.3. Under the same assumptions as in Lemma 3.2, let ̂𝑢∈𝑋 be a critical point of 𝜑𝜆. Then, ̂𝑢∈𝐶1,𝛼(Ω) and ̂𝑢 is a nonnegative solution of problem (1.3), where 0<𝛼<1.

Proof . We shall follow some of the arguments in [3]. As mentioned earlier, if ̂𝑢 is a critical point of 𝜑𝜆, then it is shown in [3] that ̂𝑢 is a solution of the differential inclusion ||||−div∇̂𝑢𝑝−2∇̂𝑢∈𝜕𝐺(̂𝑢)a.e.inΩ.(3.12) Since 𝑔 is only discontinuous at 𝑠=0, the above differential inclusion reduces to an equality, except possibly on the subset 𝐴⊂Ω where ̂𝑢=0. Since 𝑓 is subcritical, using the 𝐶1,𝛼 regularity results for quasilinear elliptic equations with 𝑝-growth condition (see, for example, [6]), we have ̂𝑢∈𝐶1,𝛼(Ω⧵𝐴). And, in this latter case, −div(|∇̂𝑢|𝑝−2∇̂𝑢) takes on values in the bounded interval [−𝑎,0]. Therefore, by the Michael selection theorem (see Theorem  1.2.5 of [5]), we see that −Δ𝑝∶𝑢↦[−𝑎,0] admits a continuous selection. Using the 𝐶1,𝛼 regularity results for quasilinear elliptic equations with 𝑝-growth condition again, it follows that ̂𝑢∈𝐶1,𝛼(Ω), 0<𝛼<1.
Next, using a Morrey-Stampacchia theorem [7, Theorem  3.2.2, page 69], we have that −div(|∇̂𝑢|𝑝−2∇̂𝑢)=0 a.e. in 𝐴. Therefore, since we defined 𝑔(0)=0, it follows that ||||−div∇̂𝑢𝑝−2∇̂𝑢=𝑔(̂𝑢)a.e.inΩ.(3.13) Replacing the inclusion (3.2) on ̂𝑢, we conclude that ̂𝑢∈𝐶1,𝛼(Ω) is a solution of (1.3). Finally, recalling that 𝑔(𝑠)=0 for 𝑠≤0, it is clear that ̂𝑢≥0. The proof of Lemma 3.3 is complete.

Lemma 3.4. Assume either (F1) or (F3), (F4). Then 𝜑𝜆 satisfies the nonsmooth (𝑃𝑆)𝑐-condition at every 𝑐∈ℝ.

Proof. Let {𝑢𝑛}𝑛≥1⊆𝑋 be a sequence such that |𝜑𝜆(𝑢𝑛)|≤𝑐 for all 𝑛≥1 and 𝑚(𝑢𝑛)→0 as 𝑛→∞. In the superlinear and subcritical case, from (F4), we have 𝑐≥𝜑𝜆𝑢𝑛=1𝑝Ω||∇𝑢𝑛||𝑝𝑑𝑥−𝜆Ω𝐺𝑢𝑛≥‖‖𝑢𝑑𝑥𝑛‖‖𝑝𝑝−𝜆𝜃Ω𝜉𝑢𝑛,𝑢𝑛≥1𝑑𝑥−𝑐𝑝−1𝜃‖‖𝑢𝑛‖‖𝑝+Ω1𝜃‖‖𝑢𝑛‖‖𝑝𝜉𝑢−𝜆𝑛,𝑢𝑛≥1𝑑𝑥−𝑐𝑝−1𝜃‖‖𝑢𝑛‖‖𝑝−𝜆𝜃‖𝜉‖𝑋∗‖‖𝑢𝑛‖‖−𝑐,(3.14) where 𝜉(𝑢𝑛)∈𝜕𝐺(𝑢𝑛). Hence {𝑢𝑛}𝑛≥1⊆𝑋 is bounded.
Thus by passing to a subsequence if necessary, we may assume that 𝑢𝑛⇀𝑢 in 𝑋 as 𝑛→+∞. We have 𝐽𝑢𝑛,𝑢𝑛−−𝑢Ω𝜉𝑢𝑛𝑢𝑛−𝑢𝑑𝑥≤𝜀𝑛‖‖𝑢𝑛‖‖−𝑢(3.15) with 𝜀𝑛↓0, where 𝜉(𝑢𝑛)∈𝜕Ψ(𝑢𝑛) and ∫𝐽=(1/𝑝)Ω|∇𝑢𝑛|𝑝𝑑𝑥. From (F3) and Chang [4] we know that 𝜉(𝑢𝑛(𝑥))∈𝐿(𝑝∗−1)′(Ω) ((𝑝∗−1)=(𝑝∗−1)/(𝑝∗−2)). Since 𝑋 is embedded compactly in 𝐿𝑝∗−1(Ω), we have that 𝑢𝑛→𝑢 as 𝑛→∞ in 𝐿𝑝∗−1(Ω). So using the Hölder inequality, we have Ω𝜉𝑛(𝑢𝑥)𝑛−𝑢𝑑𝑥⟶0as𝑛⟶+∞.(3.16)
Therefore, we obtain that lim𝑛→∞sup⟨𝐽′(𝑢𝑛),𝑢𝑛−𝑢⟩≤0. But we know that 𝐽′ is a mapping of type (𝑆+). Thus we have 𝑢𝑛⟶𝑢in𝑋.(3.17) Using the similar method, we can more easily get the nonsmooth (PS)𝑐-condition in the case of sublinear.

Remark 3.5. Note that we cannot directly apply the results of [4] because the operator −Δ𝑝 is not self-adjoint linear.

Lemma 3.6. Under assumptions (F0), (F1), and (F2), let Ω=𝐵𝑅⊂ℝ𝑁 with 𝑁≥2, and let 𝑢∈𝐶1(𝐵𝑅) be a radially symmetric, nonincreasing function such that 𝑢≥0 and 𝑢 is a minimizer of 𝜑𝜆(𝑢) with 𝜑𝜆(𝑢)=𝑚<0. Then, 𝑢 does not vanish in 𝐵𝑅; that is, 𝑢(𝑥)>0 for all 𝑥∈𝐵𝑅.

Proof. Since 𝑔 is discontinuous at zero, we note that the conclusion does not follow directly from uniqueness of solution for the Cauchy problem with data at 𝑟=𝑅 (in fact, writing 𝑢=𝑢(𝑟), 𝑟=|𝑥|, we may have 𝑢(𝑅)=𝑢′(𝑅)=0 and 𝑢≢0).
Now, since 𝑢≢0 by assumption, 𝑅0∶=inf{𝑟≤𝑅∶𝑢(𝑠)=0for𝑟≤𝑠≤𝑅} satisfies 0<𝑅0≤𝑅. If 𝑅0=𝑅 there is nothing to prove in view of the fact that 𝑢 is non-increasing. On the other hand, if 𝑅0<𝑅 then 𝑢′(𝑅0)=0 and 𝑢(𝑟)>0 for 𝑟∈[0,𝑅0). It is not hard to prove that this contradicts that 𝑢 is a minimizer of 𝜑𝜆. Indeed, if 𝑅0<𝑅then 𝜑𝜆(1𝑢)=𝑝𝐵𝑅0||||∇𝑢𝑝𝑑𝑥−𝜆𝐵𝑅0𝐺(𝑢)𝑑𝑥=𝑚<0.(3.18) A simple calculation shows that the rescaled function 𝑣(𝑟)=𝑢(𝑅0𝑟/𝑅)∈𝑊01,𝑝(𝐵𝑅)∩𝐶1(𝐵𝑅) satisfies 𝜑𝜆(𝑣)=𝛿𝑝−𝑁1𝑝𝐵𝑅0||||∇𝑢𝑝𝑑𝑥−𝛿−𝑝𝜆𝐵𝑅0𝐺(𝑢)𝑑𝑥,(3.19) where 𝛿∶=𝑅0/𝑅 is less than 1. Therefore, since we are assuming 1<𝑝≤𝑁, we would reach the contradiction 𝜑𝜆(𝑣)<𝑚.

Remark 3.7. Note that the condition (F2) is necessary because of which guarantee 𝐺 can have positive values.

Proof of Theorem 1.1. We observe that the functional 𝜑𝜆 is weakly lower semicontinuous on 𝑋. Moreover, the sublinearity assumption (F1) on 𝑔(𝑢) implies that 𝜑𝜆 is coercive. Therefore, the infimum of 𝜑𝜆 is attained at some 𝑢𝜆: inf𝑢∈𝑋𝜑𝜆(𝑢)=𝜑𝜆𝑢𝜆.(3.20) And, in view of Lemma 3.3, 𝑢𝜆∈𝐶1,𝛼(Ω) is a nonnegative solution of (1.3). We now claim that 𝑢𝜆 is nonzero for all 𝜆>0 large.

Claim 1. There exists Λ>0 such that 𝜑𝜆(𝑢𝜆)<0 for all 𝜆≥Λ.
In order to prove the claim it suffices to exhibit an element 𝑤∈𝑋 such that 𝜑𝜆(𝑤)<0 for all 𝜆>0 large. This is quite standard considering that 𝐺(𝛿)>0 by (F2). Indeed, letting Ω𝜀∶={𝑥∈Ω∶dist(𝑥,𝜕Ω)>𝜀} for 𝜀>0 small, define 𝑤 so that 𝑤(𝑥)=𝛿 for 𝑥∈Ω𝜀 and 0≤𝑤(𝑥)≤𝛿 for 𝑥∈Ω⧵Ω𝜀. Then 𝜑𝜆1(𝑤)=𝑝‖𝑤‖𝑝−𝜆Ω𝜀𝐺(𝑤)𝑑𝑥+Ω⧵Ω𝜀≤1𝐺(𝑤)𝑑𝑥𝑝‖𝑤‖𝑝Ω−𝜆𝐺(𝛿)meas𝜀−𝑐(1+𝛿𝑝)measΩ⧵Ω𝜀,(3.21) where we note that the expression in the above parenthesis is positive if we choose 𝜀>0 sufficiently small. Therefore, there exists Λ>0 such that 𝜑𝜆(𝑤)<0 for all 𝜆≥Λ, which proves the claim.
On the other hand, when Ω=𝐵𝑅, let 𝑢∗𝜆 denote the Schwarz symmetrization of 𝑢𝜆, namely, 𝑢∗𝜆 is the unique radially symmetric, nonincreasing, nonnegative function in 𝑋 which is equi-measurable with 𝑢𝜆. As is well known, 𝐵𝑅𝐺𝑢∗𝜆𝑑𝑥=𝐵𝑅𝐺𝑢𝜆𝑑𝑥,(3.22) and ‖𝑢∗𝜆‖≤‖𝑢𝜆‖ (the Faber-Krahn inequality; see [8]), so that 𝜑𝜆(𝑢∗𝜆)≤𝜑𝜆(𝑢𝜆). Therefore, we necessarily have 𝜑𝜆(𝑢∗𝜆)=𝜑𝜆(𝑢𝜆) and may assume that 𝑢𝜆=𝑢∗𝜆. Moreover, 𝑢𝜆>0 in Ω by Lemma 3.6. Therefore, 𝑢𝜆 is a positive solution of both problems (1.1) and (1.3).
Next, we recall that 𝑢=0 is a strict local minimum of 𝜑𝜆 by Lemma 3.2. Therefore, since 𝜑𝜆 satisfies the nonsmooth Palais-Smale condition by Lemma 3.4, we can use the minima 𝑢=0 and 𝑢=𝑢𝜆 of 𝜑𝜆 to apply the nonsmooth mountain pass theorem and conclude that there exists a second nontrivial critical point 𝑣𝜆 with 𝜑𝜆(𝑣𝜆)>0. Again, 𝑣𝜆 is a nonnegative solution of problem (1.3) in view of Lemma 3.3. In addition, when Ω=𝐵𝑅, arguments similar to above passage show that we may assume 𝑣𝜆=𝑣∗𝜆. The proof of Theorem 1.1 is complete.

Proof of Theorem 1.2. As is well-known, the AR condition (F4) readily implies the existence of an element 𝑒𝜆∈𝑋 such that 𝜑𝜆(𝑒𝜆)≤0. On the other hand, Lemma 3.2 says that 𝑢=0 is a (strict) local minimum of 𝜑𝜆 and Lemma 3.4 says that 𝜑𝜆 satisfies nonsmooth (PS)𝑐 for every 𝑐∈ℝ. Therefore, an application of the nonsmooth mountain pass theorem stated in section 2 yields the existence of a critical point 𝑣𝜆 such that 𝜑𝜆𝑣𝜆>0.(3.23) In particular, 𝑣𝜆≠0, and it follows that 𝑣𝜆 is a nonnegative solution of problem (1.3) by Lemma 3.3. As in the proof of Theorem 1.1, we may assume that 𝑣𝜆=𝑣∗𝜆 in the case of a ball Ω=𝐵𝑅.
Finally, still in the case of a ball Ω=𝐵𝑅, we claim that there exists 𝜆∗>0 such that, for all 0<𝜆<𝜆∗, 𝑣𝜆=𝑣∗𝜆 is a positive solution of problem (1.3) (hence of problem (1.1)) having negative normal derivative on 𝜕𝐵𝑅. Indeed, if that is not the case, then, for any given 𝜆>0, we can find 0<𝜇=𝜇(𝜆)<𝜆 such that the nonnegative solution 𝑣𝜇=𝑣∗𝜇 of problem (1.1) with 𝜆=𝜇 obtained a bove satisfies 𝑣𝜇(𝑟)>0for𝑟∈0,𝑅0,𝑣𝜇𝑅0=0,𝑣𝜇𝑅(𝑟)=0for𝑟∈0,,𝑅(3.24) for some 0<𝑅0≤𝑅. Therefore, the rescaled function 𝑤𝜇(𝑟)∶=𝑣𝜇(𝑅0𝑟/𝑅) is a positive solution of (1.1) with 𝜆=𝜇0 (again in the ball 𝐵𝑅), with 𝜇0∶=𝜇𝑅𝑝0/𝑅𝑝≤𝜇. This shows that we can always construct a decreasing sequence 𝜇𝑛>0 satisfying alternative (ii) of Theorem 1.2, in case alternative (i) does not hold.

Acknowledgments

The author is very grateful to an anonymous referee for his or her valuable suggestions. Research supported by NSFC (no. 11061030, no. 10971087).