Existence And Multiplicity Of Solutions For The Fractional Schr?dinger Equation With Magnetic Fields

In this thesis,we study firstly the following critical fractional Schr?dinger equations with magnetic field?2s(-?)A/?su+V(x)u=?f|u|u+|u|2s*-2u in RN,where ? and ? are positive parameters,s?(0,1),(-?)As denotes the fractional magnetic Laplacian operator of order s,N>2s;2s*=2N/N-2s is called fractional Sobolev critical exponent.V:RN?R and A:RN?RN are continuous electric and magnetic potentials respectively and f:R?R is a subcritical nonlinearity.Under some suitable assumptions,firstly using the methods of variation and Nehari manifold,we prove an existence of ground state solution for the above problem when ? and ? change,then by applying the Ljusternick-Schnirelmann theory and topological method,we discuss the relationship between the number of nonnegative solutions for the above problem and the topology of the set M consisting of all the smallest elements of V,and so we obtain the multiplicity of nonnegative solutions for the above equation for all sufficiently large ? and small ?.Finally,we discuss the subcritical question with magnetic field?2s(-?)A/?su+V(x)u=?f(|u|)u in RN,Under the condition that the nonlinearity satisfies a different assumption,we employ the Morse theory to obtain a multiplicity result of nontrivial solutions for the above equation for small ?.