The Existence And Multiplicity Of Solutions To Several Types Of Nonlinear Fractional Differential Equations

In this paper,we investigate the existence and multiplicity of solutions for some classes of nonlinear fractional differential equations by using variational method and critical point theory.The paper consists of six chapters and it is organized as follows.In Chapter 1,we introduce the research background,research significance,re-search status of the fractional differential equations and main works of this paper,and meanwhile,give some basic knowledge needed in this paper.In Chapter 2,we study the following nonlinear fractional Kirchhoff equation(a + b?_R³|?-??^s/2u|²dx)?-??^su + V?x?u = f?u?,x ? R3,where a>0 and b?0,s ??3/7,1?.When the potential V and the nonlinear term f satisfy some specified conditions,the existence and multiplicity of a positive solution,a negative solution and sign-changing solutions are obtained by using the mountain pass lemma and the method of invariant sets of descending flow.In Chapter 3,we consider the following nonlinear fractional Kirchhofi equation(a+??_R^N|?—??^s/2u|²dx)?—??^su + V?x?u = f?x,u?+ w?x?|u|^q-²u,x ?RN,where N>2s.a>0,??0 is a parameter.The potential V.the nonlinear ter-m f and the perturbed term w satisfy some conditions.A_s N = 3.s ??3/4,1?,the existence of two positive solutions and two negative solutions is obtained by the using the Ekeland s variational principle and the mountain pass lemma.More-over,as N>2s,s??0,1?,when the nonlinear term f does not satisfy the usual Ambrosetti-Rabinowitz condition,the existence of two positive solutions and two negative solutions is obtained by using the variational method,the iterative trick,the Pohozaev identy and the Ekeland s variational principle.In Chapter 4,we investigate the following nonlinear fractional Schrodinger-Poisson system???where s ??3/4,1?and t ??0,1?.When the potential V and the nonlinear term f satisfy some conditions,the existence and multiplicity of a positive solution,a negative solution and sign-changing solutions are obtained by means of the mountain pass lemma and the method of invariant sets of descending flow.In Chapter 5,we discuss the following fractional boundary value problem??? where ??(1/2,1],??? and ??? stand for the left and right Riemann-Liouville frac-tional derivatives of order a respectively.Under some suitable conditions on a?t?and f,the existence of infinitely many nontrivial high and small energy solutions is obtained via the variant fountain theorems.Chapter 6 is concerned with the following nonlinear impulsive fractional bound-ary value problem???where ??(1/p,1],p>1,?_p?s?=|s|^p-2s,??? represents the right Riemann-Liouville fractional derivative of order a and ??? represents the left Caputo fractional derivative of order ?,0 = t₀<t₁<… t_m+1= T and???When the impulse function Ij and the nonlinear term f satisfy some conditions,the existence of infinitely many nontrivial high and small energy solutions is obtained by using the variant fountain theorems.