Concentration Compactness And Global Compactness On Nonlocal Elliptic Partial Differential Equations

In this paper,we mainly study the existence and multiplicity of solutions to some nonlocal elliptic equations,including the Kirchhoff equation on bounded domain,the p-Kirchhoff equation,Kirchhoff equation with Hartree-type nonlinearity in R3 and the Choquard equation on bounded domain.This paper consists of six chapters.In Chapter 1,we introduce the background of the related problems and give a review on current research.In Chapter 2,we present some preliminary results which will be used throughout the paper.In Chapter 3,we study the following Kirchhoff equation?—(a + b??|?u|2dx)?u = f(x,u)+ u|u| in ?,u = 0 on??,where a,b?0,?(?)R3 is a bounded smooth domain,? is a positive parameter and f:?×R?R is a Caratheodory function satisfying some further conditions.We obtained the multiplicity of the solutions by concentration-compactness principle and symmetry mountain pass theorem.In Chapter 4,we consider the following p-Kirchhoff equation?—?a + b(??|?u|pdx)1/p-1??pu= ?uq-1+up*-1,x??,u? 0,x??,u = 0,x???,where a,?? 0,b?0,?(?)RN is a smooth bounded domain,1?p?N,1?q?p*= Np/(N—p),p*is the critical Sobolev exponent and ?p is p-Laplacian operator.By establishing a global compactness result,we obtain the existence and multiplicity of solutions under different assumptions of parameters.In Chapter 5,we study the Kirchhoff equation with Hartree-type nonlinearity—(a + b?R3??u?2)?u + V(x)u=(I?*?u?p)?u?p-2u,in R3,where a,b?0,V:R3 ? R is a potential function and I?,is a Riesz potential of order??(0,3).Under certain assumptions on V(x),we prove our problem has a ground state for all(3 + ?)/3 ? p ? 3 + ? by Pohozaev manifold methods.In Chapter 6,we study the following Choquard type equation-?u = ?u+(?? |u(y)|2*?/|x-y|N-? dy)?u?2*?-2u,in,where ?(?)RN is a smooth bounded domain,??0,N?3,0???N and 2*? =N+?/N-2 is a critical exponent.We establish a global compactness result to Choquard equation on bounded domain,which is a nonlocal counterpart of Struwe's result.