On the planar Schrodinger-Poisson system with the axially symmetric potential

In this paper, we develop some new variational and analytic techniques to prove that the following planar Schrodinger-Poisson system {-Delta u + V(x)u + phi u = f(u), x is an element of R-2, Delta phi = u(2), x is an element of R-2, admits a nontrivial solution and a ground state solution possessing the least energy in the axially symmetric functions space, where V(x) is axially symmetric. Our results improve and extend the ones in the case V = 1 and f (u) = vertical bar u vertical bar(p-2)u with 2 < p < 6. In particular, we use the assumption that 2V (x) + del V(x) . x is bounded from below instead of the usually one that lim(vertical bar x vertical bar ->infinity) V (x) = 1. Moreover, V(x) is even admitted to be unbounded. (C) 2019 Elsevier Inc. All rights reserved.