Existence of solutions for a class of quasilinear elliptic equations involving the p-Laplacian

Abstract

This paper is concerned with the existence of solutions for the quasilinear elliptic equations − Δ p u − Δ p ( | u | 2 α ) | u | 2 α − 2 u + V ( x ) | u | p − 2 u = | u | q − 2 u , x ∈ R N , where α ≥ 1 , 1<p<N, p ∗ = Np / ( N − p ) , Δ p is the p-Laplace operator and the potential 0 $ ]]> V ( x ) > 0 is a continuous function. In this work, we mainly focus on nontrivial solutions. When 2 αp < q < p ∗ , we establish the existence of nontrivial solutions by using Mountain-Pass lemma; when q ≥ 2 α p ∗ , by using a Pohozaev type variational identity, we prove that the equation has no nontrivial solutions.