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

This paper is concerned with the following planar Schrödinger-Poisson system { − Δ u + V (x) u + ϕ u = f (x , u) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where V ∈ C (R 2 , [ 0 , ∞)) is axially symmetric and f ∈ C (R 2 × 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 ϕ 2 , u (x) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity f (x , u). [ABSTRACT FROM AUTHOR]