Existence and non-existence of solutions for p-Laplacian equations with decaying cylindrical potentials

Abstract

In this paper we deal with the problem−Δpu+|u|p⁎(s)−2u|y|s=|u|m−2u,u∈D1,p(RN;R) where 1<p<N, x=(y,z)∈Rk×RN−k, Δp is p-Laplacian operator, p⁎(s)=p(N−s)N−p and p⁎=pNN−p. Combining a version of the concentration compactness result by Solimini, Hardy–Sobolev type inequality with the Mountain Pass Theorem, existence of non-trivial solutions is obtained. Decay properties of these solutions are showed by applying Vassilev results. Pohozaev type identities are established in order to get non-existence results.