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