Existence of Multiple Solutions for P-Laplacian Problems Involving Critical Exponents and Singular Cylindrical Potential

Mohammed El Mokhtar Ould

doi:10.4172/2168-9679.1000236

Abstract

In this paper, we establish the existence of multiple solutions for p-Laplacian problems involving critical exponents and singular cylindrical potential, by using Ekeland’s variational principle and mountain pass theorem without Palais-Smale conditions.

Highlights

The aim of this paper is to establish the existence and multiplicity of solutions to the following quasilinear elliptic problem ( ) λ,μ −∆ pu−= μ y − p u p−2 u u ∈ 1p ( N ), h ( y) y −s u q−2 u + λ g ( x) in N, y ≠ 0( ) Where ∆ pu= div ∇u p−2 ∇u,1< p < k, k and N are integers with N>p, p ≤ k N, =N × N
( ) The problem λ,μ has been studied by Bouchekif and Matallah in [2], by using Ekeland.s variational principle and mountain pass theorem, they established the existence of two nontrivial solutions when 0 < μ ≤ μN, λ ∈(0, Λ∗ ), where Λ∗ is a positive constant and under sufficient conditions on functions g and h
We prove the existence of at least two distinct critical points of Iλ,μ

Summary

Introduction

The aim of this paper is to establish the existence and multiplicity of solutions to the following quasilinear elliptic problem ( ) λ,μ. ( ) The problem λ,μ has been studied by Bouchekif and Matallah in [2], by using Ekeland.s variational principle and mountain pass theorem, they established the existence of two nontrivial solutions when 0 < μ ≤ μN , λ ∈(0, Λ∗ ), where Λ∗ is a positive constant and under sufficient conditions on functions g and h. In [11], Xuan studied the multiple weak solutions for p-Laplace equation with singularity and cylindrical symmetry in bounded domains. They only considered the equation with sole critical Hardy-Sobolev term. We start by recalling the following definition and properties from the paper [6]

The first inequality that we need is the Hardy inequality

Existence of a local minimizer

Existence of mountain pass type solution