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

This paper is concerned with the following planar Schrodinger-Poisson system {-Delta u + V(x)u + phi u = f(x, u), x is an element of R-2, Delta phi = u(2), x is an element of R-2, where V is an element of C(R-2, [0, infinity)) is axially symmetric and f is an element of C(R-2 x 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. Poincare Anal. Non Lineaire, 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 phi(2,u)(x) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity f(x, u). (C) 2020 Elsevier Inc. All rights reserved.