Saddle solutions for the planar Schrödinger–Poisson system with exponential growth.

In this paper, we are interested in the following planar Schrödinger–Poisson system 0.1 - Δ u + a (x) u + 2 π ϕ u = | u | p - 2 u e α 0 | u | γ , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where p > 2 , α 0 > 0 and 0 < γ ≤ 2 , the potential a : R 2 → R is invariant under the action of a closed subgroup of the orthogonal transformation group O(2). As a consequence, we obtain infinitely many saddle type nodal solutions for equation (0.1) with their nodal domains meeting at the origin if 0 < γ < 2 and p > 2 . Furthermore, in the critical case γ = 2 and p > 4 , we prove that equation (0.1) possesses a positive solution which is invariant under the same group action. [ABSTRACT FROM AUTHOR]