On the quasilinear elliptic problem with a critical Hardy–Sobolev exponent and a Hardy term

doi:10.1016/j.na.2007.08.022

Nonlinear Analysis: Theory, Methods & Applications

Volume 69, Issue 8, 15 October 2008, Pages 2432-2444

https://doi.org/10.1016/j.na.2007.08.022 Get rights and content

Abstract

In the present paper, a quasilinear elliptic problem with a critical Sobolev exponent and a Hardy-type term is considered. By means of a variational method, the existence of nontrivial solutions for the problem is obtained. The result depends crucially on the parameters $p, t, s, λ$ and $μ$ .

Section snippets

Introduction and the main results

In this paper, we consider the elliptic equation ${\begin{cases} - Δ_{p} u - μ \frac{{| u |}^{p - 2} u}{{| x |}^{p}} = \frac{{| u |}^{p^{*} (t) - 2}}{{| x |}^{t}} u + λ \frac{{| u |}^{p - 2}}{{| x |}^{s}} u, & x \in Ω, \\ u = 0, & x \in \partial Ω, \end{cases}$ where $Ω \subset R^{N} (N \geq 3)$ is a smooth bounded domain containing the origin 0, $- Δ_{p} u = - div ({| \nabla u |}^{p - 2} \nabla u), 1 < p < N, 0 \leq μ < \bar{μ} ≜ {(N - p)}^{p} / p^{p}, λ > 0, 0 \leq s, t < p, p^{*} (t) ≜ p (N - t) / (N - p)$ is the critical Hardy–Sobolev exponent.

We employ $D^{1, p} (Ω)$ to denote the closure of $C_{0}^{\infty} (Ω)$ with respect to the norm ${(\int_{Ω} {| \nabla \cdot |}^{p} d x)}^{1 / p}$ . The function $u \in D^{1, p} (Ω)$ is said to be a solution of the problem (1.1) if $u$ satisfies $\int_{Ω} ({| \nabla u |}^{p - 2} \nabla u \cdot \nabla v - μ \frac{{| u |}^{p - 2} u v}{{| x |}^{p}} - | u$

Preliminary results

In this section, we will establish several preliminary lemmas. We remark that $D^{1, p} (Ω) = H \oplus E$ , where $H = 〈 ϕ_{1} 〉 ≜ span (ϕ_{1}), ϕ_{1} > 0$ is the eigenfunction corresponding to the first eigenvalue $λ_{1}$ and $E$ is defined as in (1.4).

Lemma 2.1

Let $λ_{1}$ and $λ^{*}$ be defined as in(1.3), (1.5). Then $λ_{1} < λ^{*}$ .

Proof

The proof follows the same lines as that of Lemma 2 in [2].

To the contrary, we assume that $λ_{1} = λ^{*}$ . Then there exists ${u_{k}} \subset E$ such that $‖ u_{k} ‖ = 1$ and $λ_{1} \int_{Ω} \frac{{| u_{k} |}^{p}}{{| x |}^{s}} \to 1$ . By rescaling and setting $v_{k} = u_{k} {(\int_{Ω} \frac{{| u_{k} |}^{p}}{{| x |}^{s}})}^{- 1}$ we have $\int_{Ω} \frac{{| v_{k} |}^{p}}{{| x |}^{s}} = 1$ and $‖ v_{k}$

Proofs of the theorems

In this section, we give the proofs of Theorem 1.1, Theorem 1.2, Theorem 1.3.

Proof of Theorem 1.1

We prove that (2.6) holds for $ε$ small. To the contrary, we assume that $sup_{v \in Q_{m}^{ε}} J (v) \geq \frac{p - t}{p (N - t)} {(A_{μ, t})}^{\frac{N - t}{p - t}}, \forall m \in N, \forall ε > 0 .$ As the set ${v \in Q_{m}^{ε}; J (v) \geq 0}$ is compact, the supremum in (3.1) is attained. Then for all $ε > 0$ , there exists $w_{ε} \in H_{m}^{-}$ and $τ_{ε} \geq 0$ such that $J (v_{ε}) = sup_{v \in Q_{m}^{ε}} J (v) \geq \frac{p - t}{p (N - t)} {(A_{μ, t})}^{\frac{N - t}{p - t}},$ where $v_{ε} ≜ w_{ε} + τ_{ε} u_{ε}$ . Thus $\frac{1}{p} {‖ v_{ε} ‖}^{p} - \frac{λ}{p} \int_{Ω} \frac{{| v_{ε} |}^{p}}{{| x |}^{s}} - \frac{1}{p^{*} (t)} \int_{Ω} \frac{{| v_{ε} |}^{p^{*} (t)}}{{| x |}^{t}} \geq \frac{p - t}{p (N - t)} {(A_{μ, t})}^{\frac{N - t}{p - t}} .$ According to Lemma 2.5 the sequences ${τ_{ε}} \subset R^{+}$

Acknowledgement

This work was supported by the National Natural Science Foundation of China and the Science Foundation of SEAC of China.

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On the quasilinear elliptic problem with a critical Hardy–Sobolev exponent and a Hardy term

Abstract

Section snippets

Introduction and the main results

Preliminary results

Proofs of the theorems

Acknowledgement

J. Differential Equations

J. Differential Equations

J. Differential Equations

Nonlinear Anal.

Existence and nonexistence for quasilinear equations involving the p-laplacian

Boll. Unione Mat. Ital.

Some results on p-Laplace equations with a critical growth term

Differential Integral Equations

Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy–Sobolev exponent

Nonlinear Anal.

First order interpolation inequality with weights

Compos. Math.

Existence and nonexistence for quasilinear equations involving the $p$ -laplacian

Some results on $p$ -Laplace equations with a critical growth term