Semiclassical states for Dirac-Klein-Gordon system with critical growth

Abstract

In this paper, we study the following critical Dirac-Klein-Gordon system in R3:{iε∑k=13αk∂ku−aβu+V(x)u−λϕβu=P(x)f(|u|)u+Q(x)|u|u,−ε2Δϕ+Mϕ=4πλ(βu)⋅u, where ε>0 is a small parameter, a>0 is a constant. We prove the existence and concentration of solutions under suitable assumptions on the potential V(x),P(x) and Q(x). We also show the semiclassical solutions ωε with maximum points xε concentrating at a special set HP characterized by V(x),P(x) and Q(x), and for any sequence xε→x0∈HP,vε(x):=ωε(εx+xε) converges in H1(R3,C4) to a least energy solution u of{i∑k=13αk∂ku−aβu+V(x0)u−λϕβu=P(x0)f(|u|)u+Q(x0)|u|u,−Δϕ+Mϕ=4πλ(βu)⋅u.