Four Nontrivial Solutions for Kirchhoff Problems with Critical Potential, Critical Exponent and a Concave Term

Mohammed El Mokhtar Ould El Mokhtar

doi:10.4236/am.2015.614198

Abstract

In this paper, we consider the existence of multiple solutions to the Kirchhoff problems with critical potential, critical exponent and a concave term. Our main tools are the Nehari manifold and mountain pass theorem.

Highlights

In this paper, we consider the multiplicity results of nontrivial solutions of the following Kirchhoff problem Lμ,bu = h u 4 u +λ f u q−2 u in Ω,= u 0 on ∂Ω (1.1) ( ( ) )( ) where∫ Lμ,a,bv :=− a + b∇v 2 + μ x −2 v2 dx Ω∆v + μ x −2 v, Ω is a smooth bounded domain of
As for in nitely many solutions, we refer readers to [8] [9]. He and Zou [10] considered the class of Kirchhoff type problem when g ( x;u) = λ f ( x;u) with some conditions and proved a sequence of a.e. positive weak solutions tending to zero in L∞ (Ω)
In the last Section, we prove the Theorem 3

Summary

Introduction

We consider the multiplicity results of nontrivial solutions of the following Kirchhoff problem. As for in nitely many solutions, we refer readers to [8] [9] He and Zou [10] considered the class of Kirchhoff type problem when g ( x;u) = λ f ( x;u) with some conditions and proved a sequence of a.e. positive weak solutions tending to zero in L∞ (Ω). 3(6− q) b > (4−q) there exists λ1 > 0 such that for all λ verifying 0 < λ < min (λ0 , λ1 ) the problem (1.1) has at least two positive solutions.

Preliminaries

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3