Existence and concentration results for a (p,q)-Laplacian problem with a general critical nonlinearity

Abstract

The aim of this paper is to study the following (p,q)-Laplacian problem:{−εpΔpv−εqΔqv+V(x)(|v|p−2v+|v|q−2v)=f(v) in RN,v∈W1,p(RN)∩W1,q(RN),v>0 in RN, where ε>0 is a small parameter, N≥3, 1<p<q<N, Δsv:=div(|∇v|s−2∇v), with s∈{p,q}, is the s-Laplacian operator, V:RN→R is a positive continuous potential, and f:R→R is a Berestycki-Lions type nonlinearity with critical growth. By applying suitable variational methods, we construct a localized bound state solution that concentrates around an isolated component of a positive local minimum of V as ε→0.