Existence of a nontrivial solution for a (p,q)-Laplacian equation with p-critical exponent in RN

Abstract

In this paper we prove the existence of a nontrivial solution in D1,p(RN)∩D1,q(RN) for the following (p,q)-Laplacian problem:

{−Δpu−Δqu=λg(x)|u|r−1u+|u|p⋆−2u,u(x)≥0,x∈RN, where 1<q≤p<r+1<p⋆:=NpN−p, p<N, λ>0 is a parameter, Δmu:=div(|∇u|m−2∇u) is the m-Laplacian operator and g∈Lp⋆p⋆−r−1(RN) is positive in an open set.

MSC: 35J92, 47J30.