Research On Solution And Its Properties For Nonlinear Choq Uard Equations

This dissertation aims at researching the existence of solutions for several classes of nonlinear Choquard equations by variational method.The dissertation is divided into five chapters,the main contents are as follows:In Chapter 1,we first introduce the backgrounds and research status of prob-lems,then give the main results.In Chapter 2,we are concerned with the existence of nodal solutions for the fractional Choquard equation(?)where 0<μ<2a<N,2N-μ/N-1<p<2N-μ/N-2α with α∈(1/2,1).Since fractional Laplacian is a nonlocal operator,we adopt the extension method proposed by L.Caffarelli and L.Silvestre to reformulate the problem in R+N+1.For any k∈N we prove that the equation(P1)possesses at least a least energy k-nodal solution.In Chapter 3,we study the existence of solutions for the fractional p-Laplacian Choquard equation where a E(0,1),λ>0 is a parameter,0<μ<N,N>ap,1<q<p,max{p/2,2N-μ/2N}<γ<p/22N-μ/N-αp,f∈L∞(Ω)satisfies f±≠0 and the measure of f-1(0)is positive,Ω(?)RN is bounded domain with C1 boundary.Define(?)By combining the Nehari manifold and the fibering approach,we prove the existence of ε>0 such that for all λ∈(0,λ*+ε),the problem(P2)has two nontrivial nonnegative solutions.In Chapter 4,we consider the existence,multiplicity and concentration of pos-itive solutions for quasilinear Choquard equation(?)where ε>0 is a parameter,0<μ<2<N,pN<p<4(N-μ)/N-2 with pN-max{4,3N-2μ+2/N-2} and the potential function V satisfy the condition(V)K ∈C(RN,R),0<V0:=(?)Since there has no suitable space such that the energy functional corresponding to the equation(P3)enjoys both smoothness and compactness,we adopt the dual approach to transfer the problem into a semilinear one and prove that the equation possesses a positive ground state provided ε>0 small enough.Moreover,for smallε,by using the Ljusternik-Schnirlmann theory,we prove that the number of positive solutions for equation(P3)is at least cat(A)and as ε→0,the maximum of the above solutions concentrate near A,where A is the set of minimums of potential V.In Chapter 5,we make a summary and outlook about the dissertation.