Eigenvalue Problems For Several Kinds Of Fractional Elliptic Equations

The eigenvalue problems of fractional elliptic equation have been widely applied in physics,engineering and other fields,such as plasma physics in fusion experiments and astrophysics,the petroleum reservoir simulation,the linear stability of flows in fluid mechanics and so on.This thesis mainly uses variational methods and Nehari manifolds to study the eigenvalue problems for several kinds of fractional Kirchhoff elliptic equations.The thesis consists of the following four chapters:In chapter 1,we introduce the research background and research status of eigen-value problems of fractional elliptic equations,and gives the preliminary knowledge required for this thesis.In chapter 2,we study the following eigenvalue problems for a subcritical frac-tional Kirchhoff elliptic equationswhere（?）is a bounded region with smooth boundary in R^N,N>2s,s（?）（0,1）,a>0,2<p<min{4,2_s^*},2_s^*=（2N）/（N-2s）is the fractional critical Sobolev exponent,b,λ>0are real parameters.The existence of non-trivial solutions is obtained by using the Mountain pass theorem.In chapter 3,we investigate the following elliptic boundary value Problems for a fractional Kirchhoff elliptic equation with a sign-changing weight functionwhere（?）is a bounded region with smooth boundary in R^N,N>2s,s（?）（0,1）,a>0,2<p<min{4,2_s^*},2_s^*=（2N）/（N-2s）is the fractional critical Sobolev exponent,b,λ>0are real parameters.When g（?）L^∞（（?））which changes sign,the existence of multiple non-trivial solutions is obtained by using the Mountain pass theorem and Ekeland’s variational principle.In chapter 4,we consider the following eigenvalue problem for a fractional Kirch-hoff elliptic equation with steep potential well and concave-convex nonlinearities.(a+b∫_R^N|（-△）^s/2u|²dx)（-△）^su+λV（x）u=m（x）|u|^p-2u+k（x）|u|^q-2u,x（?）R^N,where b,λ>0 is real parameter,a>0,N≥3,s（?）（0,1）,1<q<2<p<min{4,2_s^*},2_s^*=（2N）/（N-2s）is the fractional critical Sobolev exponent.Under certain assumptions of V（x）,m（x）,k（x）,the existence of multiple positive solutions is obtained by using the Ekeland’s variational principle and Nehari manifolds.