Existence and multiplicity of positive solutions for p-Laplacian elliptic equations

Zhen Peng,Guanwei Chen

doi:10.1186/s13661-016-0632-5

Abstract

We study a p-Laplacian elliptic equation with Hardy term and Hardy-Sobolev critical exponent, where the nonlinearity is $(p-1)$ -sublinear near zero and $(p^{\ast}(s)-1)$ -sublinear near infinity ( $p^{\ast}(s)=\frac{p(N-s)}{N-p}$ is the Hardy-Sobolev critical exponent). By using variational methods and some analysis techniques, we obtain the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation. To the best of our knowledge, no result has been published concerning the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation.

Highlights

By using variational methods and some analysis techniques, we obtain the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation
Introduction and main resultsIn this paper, we will study the existence and multiplicity of positive solutions for the following p-Laplacian elliptic equation: ⎧ ⎨– pu μ |u|p– |x|p u|u|p∗ (s)– |x|s u + λf (x, u),⎩u =, x ∈ \ { }, x∈∂ . ( . )Here, ⊂ RN (N ≥ ) is an open bounded domain with smooth boundary ∂ and ∈, p ∈ (, N), s ∈ [, p), λ, μ ∈ R+, pu := div(|∇u|p– ∇u) is the p-Laplacian differential operator, p∗(s) p(N –s) N –p is exponent, p∗ p∗( )
Where c and c are positive constants depending on p and N ; a(μ) and b(μ) are zeros of the function f (t) = (p – )tp – (N – p)tp– + μ, t ≥, ≤ μ < μ, satisfying b(μ); see

Summary

Introduction

By using variational methods and some analysis techniques, we obtain the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation. To the best of our knowledge, no result has been published concerning the existence and multiplicity of positive solutions for the p-Laplacian elliptic equation. Introduction and main results In this paper, we will study the existence and multiplicity of positive solutions for the following p-Laplacian elliptic equation: There were many authors [ , – ] who have studied the existence or multiplicity of solutions for elliptic equations with the operator –

Results

Conclusion