The Existence And Multiplicity Of The Solution Of Schrödinger-Poisson Equation

In this paper,we mainly consider two problems.Firstly,we consider the following generalized Sch?dinger-Poisson system:where ? > 0,? is smooth bounded domain in R~3,We will extend some of the latest achievements from different authors.In our paper,we amuse that the nonlinear term f(u)satisfies |f(u)| ? C(|u| + |u|~?),where ? ?(1,2),which case is less studied.By Ekeland's variational principle,we obtain a nontrivial solution with negative energy.And we obtain the second solution with positive energy by structure Mountain pass geometry.Since ? ?(1,2),and the ranges of ? is related to space Embedding,we will study the problem on the smooth bounded domain ? in R~3.Secondly,we consider the following Sch?dinger-Poisson system:where g + f combines concave and convex Non-linear terms.? is smooth bounded domain in R~3,? > 0.By Mountain pass theorem,we obtain a nontrivial solution with positive energy.And by dual fountain theorem,we obtain infinitely many small energy solutions.