Existence And Multiplicity Of Solutions For A Class Of Fractional Laplacian Problems With Magnetic Fields

In this thesis,we study the existence and multiplicity of solutions for a class of fractional Laplacian problems with magnetic fields.For the following fractional Laplacian problem with a magnetic field(?) where(-?)As is fractional magnetic operator,s ?(0,1),?>0,n>2s,?(?)Rn is an open bounded subset with smooth boundary.A:Rn?Rn is continuous magnetic potential,f,g?C([0,?),R).First,we discuss the situation when ?=1,g(u)=u2s*-2,namely,the problem is a critical fractional Laplacian problem with magnetic field.If f satisfies certain conditions,we discuss the bifurcation and multiplicity of nontrivial solutions of the problem through the abstract critical point theorem.Furthermore,when the nonlinear term f satisfies some appropriate conditions,the existence of a single nontrivial solution to the above critical nonlocal problem is obtained by the direct variational method.Secondly,we study the more general fractional Laplacian problem with magnetic field.Under the suitable hypothesis of f,g,we discuss the existence of three nontrivial solutions for the nonlocal problems with perturbation terms by using the abstract three critical point theorem.