Mountain-pass type solution for planar Schrödinger–Poisson systems with critical exponential growth

Abstract

In this paper,we considered the following planar Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(x,u),x∈R2,Δϕ=u2,x∈R2,where V∈C(R2,[0,∞)) is axially symmetric and f∈C(R2×R,R) has critical exponential growth in the sense of Trudinger-Moser at t=±∞. We can convert the above system to the integro-differential equation with logarithmic convolution potential. So that we prove the existence of an axially symmetric mountain-pass type solution by some new useful estimates on logarithmic convolution potential. Our results improve the ones of Chen and Tang (2020) and some other related results in literature.