Studies On Multiple Solutions And Sign-Changing Solutions For Some Nonlocal Elliptic Equations

In this doctoral dissertation,we mainly study multiple solutions and sign-changing solutions of some nonlocal elliptic partial differential equations,in order to investigate the effect of nonlocal term on the existence of such solutions in the equations.In this paper,we consider the fractional Brezis-Nirenberg problem,fractional Schrodinger equation,nonlinear Schrodinger-Poisson equation and non-local Choquard equations.The thesis consists of four chapters which are outlined in the following:In Chapter 1,we first introduce the historical background,research signifi-cance,research status and new developments of the investigated problems in this paper,and then state our work briefly and list some preliminary tools used in the proof of our main results.In Chapter 2,we consider the following fractional Brezis-Nirenberg problemwhere ?? e(0,2),N>(1 +(?))?,Q is a smooth bounded domain in RN and(-?)?/2 is the fractional Laplacian operator.Firstly,by using the ? harmonic extension technique introduced in[1],we transform this nonloca equa.tion into an equivalent local equation.Then by using the constraint method and Z2 index theory(or Krasnoselskii genus),via constructing symmetry sets,we prove that for any real number A>0,this equation possesses at least[N+1/2]pairs nontrivial solutions.Here[a]denotes the least integer bigger than or equal to number a.In Chapter 3,we study the nonexistence of least energy sign-changing solu-tions for two nonlocal equations.This first one is fractional Schrodinger equationwhere ? ?(0,2),N? 3,p ?(2,2N/N-?).By using odd Nehari manifold and energy comparing method,we obtain the nonexistence of least energy sign-changing solu-tions and least energy odd sign-changing solutions.Applying this method to the following nonlinear Schrodinger-Poisson equationwhere p ?(3,5)and A>0,we also obtain the nonexistence of least energy signn-changing solutions and least energy odd sign-changing solutions.We find that both nonlocal equations have in common the nonexistence of least energy sing-changing solutions and the nonlocal terms do not change the "double energy" property.In Chapter 4,we study the following nonlocal Choquard type equationwhere p E[2,6),q ?(1,5),?>and ?(?)R3 is smooth bounded domain.We first use the method of Nehari manifold to prove the existence of ground state solu-tions when ?,?>0.Then by using variational methods,topological degree theory,implicit function theorem,generalized Nehari method and energy comparing tech-nique,we prove that for any ?>0,when p?(2,6),?>0 or p=2,?<?1,if q? 2,then the equation admits least energy sign-changing solutions;if 1<q<2,?>0,then the equation admits no such solution.Moreover,when p = 2,A ??1 for any q ?(1,5),the equation admits ground state solutions which are least energy sign-changing solutions.Here ?1 denotes the first eigenfunction of-? under zero boundary condition.This result shows that q = 2 is the critical value for the exis-tence of least energy sign-changing solutions to the equation above,and that the presence of nonlocal term ??u|u|q-2u makes the energy functional totally different from the case ? = 0.