Quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities

Abstract

In this paper, we consider the following quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities −normalΔu+ϕu=false|ufalse|p−2ulogfalse|ufalse|2+λffalse(ufalse),0.30em0.1emin0.51emnormalΩ,−normalΔϕ−ε4normalΔ4ϕ=u2,0.30em0.1emin0.51emnormalΩ,u=ϕ=0,0.30em0.1emon0.51em∂normalΩ, where 4<p<+∞,0.1emε,λ>0 are parameters, normalΔ4ϕ=divfalse(false|∇ϕfalse|2∇ϕfalse),normalΩ⊂ℝ2 is a bounded domain, and f has exponential critical growth. By adopting the reduction argument and a truncation technique, we prove for every ε>0, the above system admits at least one pair of nonnegative solutions false(uε,λ,ϕε,λfalse) for λ>0 large. Furthermore, we research the asymptotical behavior of solutions with respect to the parameters ε and λ. The novelty of this system is the intersection among the quasilinear term, logarithmic term, and exponential critical term. These results are new and improve some existing results in the literature.