Quasi-linear Schrödinger–Poisson system under an exponential critical nonlinearity: existence and asymptotic behaviour of solutions

In this paper we consider the following quasilinear Schrödinger–Poisson system in a bounded domain in R 2 : {-Δu+ϕu=f(u)inΩ,-Δϕ-ε4Δ4ϕ=u2inΩ,u=ϕ=0on∂Ωdepending on the parameter ε> 0. The nonlinearity f is assumed to have critical exponential growth. We first prove existence of nontrivial solutions (u ε , ϕ ε ) and then we show that as ε→ 0 + , these solutions converge to a nontrivial solution of the associated Schrödinger–Poisson system, that is, by making ε= 0 in the system above.