The Existence And Multiplicity Of Sign-changing Solutions For Elliptic System With Mixed Operators

In this thesis,we consider the following Schr(?)dinger-Poisson system with mixed operator (?) where s,t∈(0,1)and 4s+2t≥3.By using the method of invariant sets of descending flow and perturbation approach,we prove the existence and multiplicity of sign-changing solutions for the system under appropriate assumptions on the potential V(x)and the nonlinear term f(u).And we show the asymptotic behavior of the ground state sign-changing solution when ε→0 and ε→∞.This thesis is divided into six chapters,the main ideas are as follows:In Chapter 1,we introduce the research background,research significance,research status at home and abroad,and the main research problems.In Chapter 2,we state some elementary knowledge about Sobolev spaces and lemmas which will be used in the thesis.In Chapter 3,when μ>4 in(AR)condition,we prove the existence and multiplicity of the sign-changing solutions by using minimax arguments in the presence of invariant sets of a descending flow for the variational formulation.In Chapter 4,when μ>2+2/(s+t)in the(AR)condition,the geometric properties in the variational structure change.We use the invariant sets of a descending flow based on perturbation method to prove the existence and multiplicity of sign-changing solutions.In Chapter 5,when ε→0 or ε→∞,the asymptotic behavior of a ground state signchanging solution is studied.In Chapter 6,we summarize the whole paper and give the prospect.