Existence And Multiplicity Of Solutions For Two Classes Of Kirchhoff-Choquard Type Equations

In recent years,the Choquard equation,as a class of nonlocal second-order elliptic partial differential equation,has been widely used in quantum mechanics,nonlinear optics and Bose-einstein condensation.The study of the existence and multiplicity of solutions of Choquard equations has become a hot topic in the field of nonlinear analysis.The main purpose of this thesis is to study the existence and multiplicity of solutions for Kirchhoff-Choquard type equations in RN.This thesis consists four chapters.In the first chapter,we give the background knowledge and significance of KirchhoffChoquard type equations,moreover,some related concepts and methods are introduced.In chapter 2,we give the common research methods and techniques of nonlinear functional analysis involved in this thesis.In chapter 3,we investigate the degenerate fractional Kirchhoff-choquard type equation with critical growth:where x∈RN,Iμ(x)=xN-μ with 0<μ<N,‖u‖s(∫∫R2N|u(x)-ei(x-y)·A(x+y/2)u(y)|2/|x-y|N+2s dxdy+∫RN|u|2dx)1/2,(-Δ)As and A are called the magnetic operator and magnetic potential,respectively.m:R0+→R0+ is a continuous Kirchhoff function,C1-function G:[0,+∞)×R2→R+ satisfy suitable conditions and p*=N+μ/N-2s.We use the principle of compactness,overcoming the compactness loss caused by critical exponents,and combine it with the limit index theorem to prove the existence and multiplicity of solutions to above equation.In chapter 4,we study Kirchhoff-Choquard equations in RN:where Δp is the p-Laplacian operator,0<μ<p<N,b,λ are some positive parameters,f:R→R is a continuous function,and the potential V:RN→R is a nonnegative continuous function.The potentials V:RN→R,is a continuous function,assume that the set of zeros of V(x)has an isolated connected branch,Ω1,Ω2,…Ωk,so that Qj inside is not empty,and ?Ωj is smooth,then for a sufficiently large penalty technique λ>0,using penalty technique,Morse iteration technique and variational method,it is proved that the above system has at least 2k-1 multi-bump solutions.We extend the results of the literature cite[71,72]to the Kirchhoff-Choquard equation,so we need to establish some new inequalities to overcome the difficulties of the two local terms.