Research On The Existence Of Positive Solutions For One Type Nonlinear Elliptic System

The existence and properties of solutions for nonlinear elliptic equations;by using the variational method has always been a hot topic for many scholars.These models are derived from fields such as semiconductor theory and nonlinear optics.Therefore,the solutions of these problems have certain the.oretical and practical signifiGance.In this thesis,the variational method is used to study the existence and properties of the solrution in the nonlinear Schroodinger-Poisson system:?-?u+V(x)u+K(x)?(x)u=?f(u),x?R3,-???K(x)u2,x?R3.We pro.ve that the system.has a positive solution by using the variational method to carefully study properties of PS sequence,where the energy of this positive solution is higher than the ground state solution.This thesis contains the following five p'arts:The first chapter mainly describes the research background,research status,research content and main result of the problem.The second chapter gives the basic knowledge and theorems involved in this paper.The third chapter mainly explores the existence of positive bound state solutions in the general nonline.ar Schrodinger-Poisson system.Using the variational method,it is proved that when the nonlinear term satisfies certain conditions,the equation has a positive bound state solution.The fourth chapter mainly explores the existence of pos.itive solutions for the supercritical nonlinear Schrodinger-Poisson system.We prove the existence of a positive solution to the problem when the nonlinearity f is subcritical at infinity and supercritical near the origin,and the potential V vanishes at infinity.The final chapter summarizes the whole thesis and looks forward to the future research direction.