Standing Wave Solutions To A Two-dimensional Schr(?)dinger-poisson System

In this paper,we mainly study the existence and some important properties of the standing wave solutions of a nonlinear Schrodinger-Poisson(SP)system.To be specific,this paper can be divided into several parts.Firstly,we can prove the ex-istence of standing wave solutions by using variational method on the L2-norm constraint,that is,we convert the existence of solutions into the existence of minimizers for the corresponding energy functional.In this process,we can finally prove the existence of minimizers by verifying the coercive conditions,weakly lower semi-continuity and some compactness conditions for the energy functional on the constraint,which means the existence of weak solutions of the system.Then,according to the normalized elliptical regularity theory,we find that this weak solution is actually a classical solution and we get the decreasing rate of the above solution at infinity.Finally,applying the moving plane method,we acknowledge that the above classical solution is radially symmetric up to translation and decreasing about the symmetric center.Since the nonlinear potential is neither bounded from above nor from below,we should think carefully how to deal with the nonlinear potential and how to choose a suitable functional space where the nonlinear potential is well-defined.According to some results of others,we decompose the difficult nonlinear potential into two relatively easy parts and choose a suitable smaller func-tional space to ensure the two nonlinear parts are well-defined in this smaller space,which can permit the energy functional is well-defined in this smaller space.Moreover,we can prove the energy functional satisfies some smoothness.