The Applications Of Critical Point Theory For A Class Of Fractional Differential Systems

In this Ph.D. thesis, by using critical point theory we deal with the existence and multiplicity of solutions for the following kind of fractional differential systems with Dirichlet boundary conditions, which come from the steady state fractional advection dispersion equations.This thesis is divided into four chapters.In Chapter1, the historic development of fractional calculus and the physical model of our investigated problem are introduced briefly. Besides, the status and the up-to-date progress for all the related problems are given, the main contents of the dissertation are outlined, some related preliminaries are recalled. At the end of this chapter, the established variational setting for our problem are also introduced.In Chapter2, by using Mountain Pass Lemma, we investigate the existence of nontrivial solutions and the minimum energy solution for the above boundary value problem when the nonlinearity F is superquadratic at infinity. By using Fountain Theorem and Symmetric Mountain Pass Lemma respectively, we establish the existence of infinitely many high energy solutions for the boundary value problem. In our results, a series of new existence conditions are obtained, which weak the (AR) condition usually needed in known results efficiently.In Chapter3, using Clark theorem in critical point theory as a main tool, we establish the existence criteria to guarantee that the above boundary value problem has at least one or infinitely many nontrivial solutions when F is subquadratic at infinity. It seems that no similar results can be found in the literature.In Chapter4, some interesting existence conditions are obtained when F is asymptotically quadratic at infinity. By applying Mountain Pass Lemma, we study the existence of nontrivial solutions for the above boundary value problem. Under the assumption that F is even, by using Dual Fountain Theorem, we obtain the boundary value problem has infinitely many low energy solutions. To the best our knowledge, similar results cannot be found in the literature in this case as same as that one when F is subquadratic.