Axially symmetric solutions for planar Schrödinger–Poisson systems with critical exponential growth and non-negative potential

Abstract

This paper is concerned with the following planar Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(u),x∈R2,Δϕ=u2,x∈R2,where V∈C(R2,[0,∞)) is axially symmetric and f∈C(R,R) has critical exponential growth in the sense of Trudinger–Moser. This system can be converted into the integro-differential equation with logarithmic convolution potential. We prove the existence of axially symmetric solutions by some new useful estimates on logarithmic convolution potential. Our result not only improves the ones of Chen–Tang (2020), but covers the zero mass case that V(x)≡0.