Existence and asymptotic behaviour of standing waves for quasilinear Schrödinger–Poisson systems in R3R3

In this work we are concerned with existence and asymptotic behaviour of standing wave solutions in the whole space R3R3 for the quasilinear Schrödinger–Poisson system−12Δu+(V+V˜)u+ωu=0,−div[(1+ε4|∇V|2)∇V]=|u|2−n∗, when the nonlinearity coefficient ε>0ε>0 goes to zero. Under appropriate, almost optimal, assumptions on the potential V˜ and the density n∗n∗ we establish existence of a ground state (uε,Vε)(uε,Vε) of the above system, for all ε   sufficiently small, and show that (uε,Vε)(uε,Vε) converges to (u0,V0)(u0,V0), the ground state solution of the corresponding system for ε=0ε=0.