Spectrum Theory Of Fractional Laplace Operator And Its Applications In Differential Equations

The dissertation mainly focuses on the spectrum theory of fractional Laplace bperators with constant sign weight. As the applications of its, we establish the global bifurcation structure of fractional Laplacian perturbation problems, and study the existence of constant sign solutions of nonlinear fractional differential gquations. The main content is concretely divided into following five chapters.Firstly, we give a survey to the development of fractional differential equa-tions, and briefly introduce the main results of this paper. At the same time, we also introduce the definitions of fractional Laplace operator, fractional derivative, integrate and some basic properties of them.Secondly, by using the Ljusternik-Schnirelmann theory, we study the eigen-values and eigenfunctions of the fractional Laplace operator. Particularly, we show that there exists a simple, isolated principal eigenvalue λ1. Furthermore, for the sake of convenience, we establish the unilateral global bifurcation theorem for fractional Laplacian perturbation problem. Under some natural hypotheses on perturbation function g, we derive that (λ1,0) is a bifurcation point of the )roblem and there are two distinct unbounded sub-continua C+ and C_, consisting of the continuum C emanating from (λ1,0). Based on the above unilateral global bifurcation result, we investigate the existence of constant sign solutions for a class of nonlinear fractional Laplace problem. The main results of spectrum theory and the existence of constant sign solutions partially improved the known esults about Servadei et al [Discrete Contin. Dyn. Syst.2013], [J. Math. Anal. Appl.2012] and Fiscella [Topol. Methods Nonlinear Anal.2014].Then, we consider the unilateral global bifurcation structure of fractional differential equation with nondifferentiable nonlinearity, and we study the bifurcation structure of above problem from trivial solution line and infinity respectively. There are two distinct unbounded sub-continua C+and C_, consisting of the continuum C em-anating from [λ1- d, λ1+d] ×{0}, and unbounded sub-continua D1 and D consisting of the continuum D emanating from [λ1- d, λ1+d]×{+∞}, where λ1 is the principal eigenvalue of corresponding linear fractional Laplace operator, and d, d are positive constants. As applications of the above bifurcation results, we let the existence of the principal half-eigenvalues of half-linear fractional differ-ential operator. Moreover, we investigate the existence of constant sign solutions for a class of fractional nonlinear problem. The main results of this chapter ex-pend the known results for classical elliptic differential equations to the fractional aplacian, there results about the fractional are new to us.Later, we give a global description of the branches of positive solutions of fractional order two point boundary value problemvhere RD0α+is the standard α∈ (1,2] order Riemann-Liouville fractional deriva-tives, and λ>0 is a real parameter. Our approach is based on topological degree and global bifurcation techniques. The main theorem of this chapter partially extends and improves the main result of Bai et al [J. Math. Anal. Appl.2005].The last one is concerned with the existence of solutions to the following ractional differential inclusionwhere 0Dx-β and xD15 are left and right Riemann-Liouville fractional integrals of )rder β∈ (0,1) respectively,0<p=1-g<1 and F:[0,1]×R→R R is locally Lipschitz with respect to the second variable. Due to the general assumption on the constants p and q, the above problem does not have variational structure. Despite that, here we study it by using nonsmooth critical point theory and iter-ative technique, and we obtain an existence result for the above problem under suitable assumptions. The result extends the case of p= q=1/2 in Teng et al [Appl. Math. Comput.20131.