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

This paper is concerned with the existence of solutions for the quasi-linear elliptic equations -delta(p)u-delta(p)(|u|(2 alpha))|u|(2 alpha-2)u + V(x)|u|(p-2)u = |u|(q-2)u, x is an element of R-N,where alpha >= 1,1 < p < N, p* = Np/(N - p), delta(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 2 alpha p < q < p*, we establish the existence of nontrivial solutions by using Mountain-Pass lemma; when q >= 2 alpha p*, by using a Pohozaev type variational identity, we prove that the equation has no nontrivial solutions.