The Existence Of Nontrivial Solution To Schr?dinger Problem With Sign-changing Potential And Logarithmic Nonlinearity

The Schr?dinger equation with logarithmic nonlinearity is an important class of nonlinear partial differential equations.Many physical phenomena can be described by the Schr?dinger problem with logarithmic nonlinearity,such as effective quantum gravity,quantum mechanics,and nuclear physics,therefore,studying such problems has great theoretical significance and practical value.This paper studies the properties of the solution of the Schr?dinger equation with logarithmic nonlinearity,it mainly uses the variational method,mountain pass theorem,Nehari manifold,and the logarithmic Sobolev inequality.The thesis consists of three sections.In Chapter 1,preface.In Chapter 2,we first study the following Schr?dinger problem with Hatree and logarithmic nonlinearitywhere ? ? R3 is a bounded domain with a smooth boundary.The potential function V satisfies the following conditions:(?)and(?),where (?).The main conclusion is as follows:Theorem 2.1.1.Suppose that(V) holds.Then the problem(0.3)has a nontrivial solution.In Chapter 3,we consider the following Schr?dinger problem with logarithmic nonlinearitywhere ? ? R3 is a bounded domain with a smooth boundary.We assume that the potential function H satisfies the following conditions:(H)H ? C(?) and changes sign in (?),satisfyingTheorem 3.1.1.Suppose that(V) and (H) hold.Then the problem(0.4)has at least two nontrivial solutions.