Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth

Abstract

This paper is concerned with 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) is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We obtain the existence of a nontrivial solution or a ground state solution of Nehari-type and infinitely many solutions to the above system under weak assumptions on V and f. Our theorems extend the results of Cingolani and Weth [Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016) 169-197] and of Du and Weth [Nonlinearity, 30 (2017) 3492-3515] and Chen and Tang [J. Differential Equations, 268 (2020) 945-976], where f(x,u) has polynomial growth on u. In particular, some new tricks and approaches are introduced to overcome the double difficulties resulting from the appearance of both the convolution ϕ2,u(x) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity f(x,u).