The Existence And Multiplicity Of Solutions For A Class Of Nonlocal Schrodinger Equation With An Integro-differential Operator

Represented by fractional Schrodinger equations of nonlocal elliptic equations are widely appeared in quantum mechanics and geometry.Apply variational methods to investigate the existence and multiplicity of nonlocal elliptic equations is a widely concerned problem in nonlinear analisis in recent years.However,the compactness of embedding of Sobolev space plays an important role.In this thesis,let G be a subgroup of O(N)and ?(?)RN is compatible with G.First,Lions vanishing lemma is generalized and we show the embedding of the G-invariant subspace of Sobolev spaces Xs(?)is compact,then we use the principle of symmetric criticality to show the existence and multiplicity of radial solutions and nonradial solutions of a class of nonlocal Schrodinger equation,which includes an integro-differential operator and more general then the fractional Schrodinger equation.Next,under the coercive condition of the potential V(x),we show the compact embedding of the subspace of nonlocal Sobolev space Xs(RN)with the weighted V(x)square integrable.Based on this,we prove the existence of the ground-state sign-changing solution(the one with the lowest energy among all the sign-changing solutions)of the above nonlocal Schrodinger equation by using the constraints on the sign-changing Nehari manifold and the existence of the infinitely many sign-changing solutions of the above nonlocal Schrodinger equation by using the method of invariant sets with descending flow.