| The nonlinear Schr(?)dinger-Poisson equation is a type of nonlinear Schr(?)dinger equations with non-local terms.It arises in semiconductor theory and has been studied extensively by domestic and foreign scholars in the past ten years.Because the Schr(?)dinger equation with logarithmic nonlinear term is widely used in quantum mechanics,quantum optics,nuclear physics and so on,this thesis mainly studies the existence and multiplicity of solutions for the logarithmic Schr(?)dinger-Poisson equation via the constrained variational method.The main research contents of this thesis include the following two parts:In the first part,this thesis studies the existence of solutions for the Schr(?)dinger-Poisson equation with logarithmic nonlinear term.Because the corresponding energy functional is not C~1 smooth,the classical critical point theory is no longer applicable.In this thesis,the existence of the reachable element of the energy functional minimization problem is obtained via using the constrained variational method and some scaling transformation techniques.Then,it is proved that the reachable element is the solution of the equation,and thus the existence result of the solutions of the equation is obtained.In the second part,this thesis further studies the multiplicity of solutions of the Schr(?)dinger-Poisson equation with logarithmic nonlinear term.In this thesis,the energy functional corresponding to the equation is first decomposed into the sum of a C~1 functional and a convex lower semicontinuous functional,and then a sequence of minimax values is defined via using ideas of Fountain theorem,and it is proved that the intersection of these minimax point sets and the set of critical point of energy functional is non-empty.Therefore,the multiplicity result of the solutions of the equation is obtained.Up to now,there are few literatures on the existence of solutions of the Schr(?)dinger-Poisson equation with logarithmic nonlinear term.This thesis mainly studies the existence and multiplicity of the solutions of the logarithmic Schr(?)dinger-Poisson equation.These results are helpful to improve the understanding on the physical model of Schr(?)dinger-Poisson equation with logarithmic nonlinear term.At the same time,it will further promote the application of nonlinear functional analysis theory in this kind of problems. |
