Solvability Of Some Classes Of Boundary Value Problems For Fractional P-Laplacian Equations

Fractional calculus is a generalization of the ordinary differentiation and integration. The fractional order models can be found to be more adequate than the integer order models in some real world problems as the fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. For example, the fractional differential equations are widely used in neurons, electrochemistry, control, and so on. The p-Laplacian equation is derived from the turbulent flow in a porous medium of mechanics, and often occurs in non-Newtonian fluid theory, nonlinear elastic mechanics, and so forth.In recent years, many scholars studied the existence and multiplicity of solutions for fractional boundary value problems, and obtained a lot of important results. Most of tools used are the nonlinear analysis methods, such as fixed point theorem, topological degree theory, and so on. But the results obtained by using critical point theory are very few, since it is often very difficult to establish the suitable function spaces and variational functionals for fractional boundary value problems.In this dissertation, we discuss the solvability of some classes of boundary value problems for fractional p-Laplacian equations. By using critical point theory and degree theory, we obtain some results on the existence of solutions and infinitely many solutions for such fractional boundary value problems. The results obtained extend and enrich some known work to some extent. This dissertation is divided into six chapters,and the detailed contents are as follows.In Chapter 1, we introduce the research significance and research status of our work, and state our main results. Moreover, some basic concepts and properties of fractional calculus are introduced.In Chapter 2, we discuss the multiplicity of solutions of Dirichlet problems for fractional p-Laplacian equation and Kirchhoff-type fractional p-Laplacian equation under the variational framework. When the nonlinearity is(p- 1)-sublinear((p2- 1)-sublinear) at infinity, we prove that such Dirichlet problems possess infinitely many nontrivial weak solutions by using the genus properties. Because the p-Laplacian operator and Kirchhoff term are nonlinear, it is difficult to verify the(PS)-condition.In Chapter 3, we discuss the solvability of Dirichlet problems for fractional pLaplacian equation and Kirchhoff-type fractional p-Laplacian equation under the variational framework. When the nonlinearity satisfy the Ambrosetti-Rabinowtiz condition,we prove that such Dirichlet problems possess at least one nontrivial weak solution by using the mountain pass theorem, and possess at least one nontrivial ground state solution by using the Nehari method. Since the Kirchhoff term is nonlinear, it is difficult to verify the Nehari manifold and the convexity of value mapping. In addition,the Ambrosetti-Rabinowtiz condition can guarantee that the nonlinearity is(p- 1)-superlinear((p2-1)-superlinear) at infinity, which is different from the(p-1)-sublinear((p2- 1)-sublinear) condition in Chapter 2.In Chapter 4, we discuss the solvability of periodic boundary value problem for fractional p-Laplacian equation under the degree theory framework. Firstly we establish a continuation theorem for the fractional p-Laplacian operator with periodic boundary conditions. Then, under the growth condition and the symbol condition, we obtain the existence result on solutions for such periodic boundary value problem by using this continuation theorem. Since the fractional p-Laplacian operator is a nonlinear operator and the Mawhin continuation theorem is only valid for linear operators, the continuation theorem established is an extension of the Mawhin continuation theorem.In Chapter 5, we discuss the solvability of some classes of boundary value problems for fractional p-Laplacian equation at resonance under the degree theory framework. When the nonlinearity satisfies the growth conditions and the symbol conditions,we obtain some results on the existence of solutions for such resonance boundary value problems by using the continuation theorem. The associated homogeneous boundary value problem of resonance boundary value problem has a nontrivial solution, so the corresponding differential operator is irreversible. Moreover, because the Mawhin continuation theorem can only deal with the linear operators, we turn the nonlinear boundary value problems into corresponding linear systems to be discussed.The main results of this dissertation and the following research work are given in Chapter 6.