Concentrated solutions to the Schrödinger–Bopp–Podolsky system with a positive potential

Abstract

Consider the Schrödinger–Bopp–Podolsky system(SBPϵ){−ϵ2Δu+(V+Kϕ)u=u|u|p−1;Δ2ϕ−Δϕ=4πKu2inR3 for sufficiently small ϵ>0, where V,K:R3→[0,∞[; p∈]1,5[ are fixed and we want to solve for u,ϕ:R3→R. Under certain hypotheses, we estimate the multiplicity of solutions in function of a critical manifold of V and we establish the existence of solutions concentrated around critical points of V.