Existence of multiple solutions for a nonhomogeneous p-Laplacian elliptic equation with critical Sobolev-Hardy exponent

Atika Matallah,Sofiane Messirdi,Sara Litimein

doi:10.5269/bspm.46206

Abstract

This paper is concerned with the existence of multiple nontrivial solutions for nonhomogeneous p-Laplacain elliptic problems involving the critical Hardy-Sobolev exponent. The method used here is based on the Nehari manifold.

Highlights

Elliptic problems involving the Hardy inequality or Hardy–Sobolev inequality has been studied by some authors either in bounded domain or in the whole space RN, see [1, 2, 6, 8,9,10,11,12] and the references therein
A natural interesting question is whether the results concerning the solutions of (P0,0) with p = 2 remain true for the problem (Pμ,s)
As in [13], we study in this paper the problem (Pμ,s) and give some positive answers

Summary

Introduction

Elliptic problems involving the Hardy inequality or Hardy–Sobolev inequality has been studied by some authors either in bounded domain or in the whole space RN , see [1, 2, 6, 8,9,10,11,12] and the references therein. When the problem (Pμ,s) has no singular term (s = μ = 0), Tarantello in [13] proved the existence of two nontrivial solutions for p = 2 and f ∈ H−1 the dual of H01 such that By the previous lemma we conclude that N = N+ ∪ N−, and we can define m+ := inf I (u) and m− := inf I (u) . Suppose that Λf > 0, we have: i) The functional I is coercive and bounded from below on N. ii) There exist m+0 < 0 such that inf u∈N

Results

Conclusion