Existence Of Solutions For Non-local Nonlinear Elliptic Problems

In this thesis,we are concerned with nonlocal elliptic problems,involving fractional Kirchhoff-type equations and Schrodinger-Poisson systems.By virtue of the variational method and a local deformation,we show the existence of positive solutions to a fractional Kirchhoff-type equation.Moreover,via a perturbation method and the method of invariant sets of descending flow,we proved the existence of sign-changing solutions to a fractional Kirchhoff-type equation and a Schrodinger-Poisson system.This thesis is divided into five chapters.It is organized as follows:In Chapter 1,we introduce the background and significance of our prob-lems.Meanwhile,the preliminary knowledge and main results of this thesis are presented.In Chapter 2,we consider the following fractional Kirchhoff-type prob-lem(?)with ?>0,??(0,1).Thanks to the variational method and a local deformation,we prove that the problem above admits positive solutions in the subcritical and critical cases.This result extends one in[1]to the fractional case.In Chapter 3,we aim to study a class of semilinear fractional Kirchhoff equations in the following form(?)where ??(3/4,1)M(t)=(?)(s)ds and m ?C(R+,R+).Via the method of invariant sets of descending flow and a perturbation approach,sign-changing solutions are obtained.In particular,the well-known Ambrosetti-Rabinowitz condition may not be required.This result extends one in[2]to the fractional case.In Chapter 4,it is dedicated to investigating the existence of sign-changing solutions to the following Schrodinger-Poisson system.(?)where ?>0.By using the method of invariant sets of descending flow,we prove that the problem above admits multiple radial sign-changing solutions in the subquadratic case as ? small.As for the quadratic case,by virtue of a perturbation argument,we show that this problem admits at least one sign-changing solution as? small.In Chapter 5,we summarize the results of this thesis and make some discussions.