Group invariant solutions for the planar Schro<spacing diaeresis>dinger-Poisson equations

This paper is concerned with the following planar Schro center dot dinger-Poisson equations -triangle u +V(x)u + (ln| <middle dot> |& lowast; |u|(p))|u|(p-2)u= f(x,u), x is an element of R-2,where p > 2 is a constant, and V(x) and f(x, u) are continuous, mirror symmetric or rotationally periodic functions. The nonlinear term f(x, u) satisfies a certain monotonicity condition and has critical exponential growth in the Trudinger-Moser sense. We adopted a version of mountain pass theorem by constructing a Cerami sequence, which in turn leads to a ground state solution. Our method has two RR new insights. First, we observed that the integral(R2 ) integral(R2) ln (|x - y|)|u(x)|(p)|u(y)|(p)dxdy is always negative R2 if u belongs to a suitable space. Second, we built a new Moser type function to ensure the boundedness of the Cerami sequence, which further guarantees its compactness. In particular, by replacing the monotonicity condition with the Ambrosetti-Rabinowitz condition, our approach works also for the subcritical growth case.