Quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities

摘要

In this paper, we consider the following quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities −Δu+ϕu=up−2ulogu2+λf(u),inΩ,−Δϕ−ε4Δ4ϕ=u2,inΩ,u=ϕ=0,on∂Ω, where 40 are parameters, Δ4ϕ=div(∇ϕ2∇ϕ),Ω⊂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 (uε,λ,ϕε,λ) 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.