The Existence And Multiplicity Of Solutions For Several Kinds Of Fractional Differential Equations

In this thesis,by using the variational method and some analysis techniques,we study the existence and multiplicity of solutions for some class of fractional differential equations.In Chapter 1,we introduce physical background and research status on frac-tional differential equations,give some basic notations,definitions and necessary theorems,then we make provision for the structure of the thesis.In Chapter 2,we study the following fractional Schr(?)dinger equation with potentialwhere ??(0,1),N>2?,the nonlinear f ?C(R~N×R,R)satisfies superlinear growth.When V(x)is periodic potential or sign-changing potential,we obtain the existence and multiplicity of solutions via the minimax methods.In Chapter 3,we consider the following fractional equation with critical exponentwhere ?(?)R~N(N>2s)is a smooth bounded domain,s ?(0,1),0<q<1,2_s~*=2N/N-2sis the fractional critical exponent.Under appropriate condition on coefficient function f(x) and the other coefficient function g ? L~?(?)is nonzero and nonnegative,we get the multiplicity of positive solutions and the existence of ground state solution by using Nehari method and Mountain Pass Lemma.In Chapter 4,we study the following Schr(?)dinger equation with non-local fractional operatorwhere ??(0,1),N>2?,(-?)_?~? denotes a non-local fractional Laplacian oper-ator with a range of scope determined by the positive function p? C(R~N;R~+),the nonlinear f?C(R~N×R~+?R)satisfies superlinear growth.When V(x)is a coercive potential,we prove the existence of a nonnegative ground state solution by using Mountain Pass Lemma.