Multiplicity and concentration properties for (p,q)-Kirchhoff non-autonomous problems with Choquard nonlinearity

Abstract

In this paper, we study the following (p,q)-Kirchhoff problem with Choquard nonlinearity:−(1+a∫RN|∇u|pdx)Δpu−(1+b∫RN|∇u|qdx)Δqu+Vε(x)(|u|p−2u+|u|q−2u)=(|x|−μ⁎F(u))f(u)inRN, where ε is a small positive parameter, a,b are positive constants, 1<p<q<N, q<2p, Δsu=div(|∇u|s−2∇u) with s∈{p,q} is the s-Laplacian, the potential V:RN→R is continuous, Vε(x)=V(εx), 0<μ<q, f is a continuous nonlinearity, and F is the primitive of f. The main result in this paper establishes multiplicity and concentration properties of positive solutions under weaker hypotheses. The proofs combine nonstandard Nehari manifold methods, penalization techniques and the Ljusternik-Schnirelmann category theory.