The nonlinear (p,q)-Schrödinger equation with a general nonlinearity: Existence and concentration

Abstract

We investigate the following class of (p,q)-Laplacian problems:{−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 continuous potential such that infRN⁡V>0 and V0:=infΛ⁡V<min∂Λ⁡V for some bounded open set Λ⊂RN, and f:R→R is a subcritical Berestycki-Lions type nonlinearity. Using variational arguments, we show the existence of a family of solutions (vε) which concentrates around M:={x∈Λ:V(x)=V0} as ε→0.