Existence Results For A Class Of Schr(?)dinger-Maxwell Equations And Sequential Fractional Differential Equations

Many nonlinear problems have their roots in geometry, astronomy, fluid and elastic mechanics, physics, chemistry, biology, control theory, image processing and economics. The main purpose of nonlinear functional analysis is to develop abstract topological and variational methods to study nonlinear problems arising in these applied subjects. Although nonlinear functional analysis is a rather recent field, initiated about one hun-dred years ago, remarkable advances have been made. Especially, in the past thirty years it has undergo rapid growth. And it has become part of the mainstream research fields in contemporary mathematical analysis. The theories and methods in nonlin-ear functional analysis stem from many areas of mathematics:ordinary differential equations, partial differential equations, the calculus of variations, dynamical systems, differential geometry, Lie groups, algebraic topology, linear functional analysis, mea-sure theory, harmonic analysis, convex analysis, game theory, optimization theory, etc. Now the main ingredients of nonlinear analysis consist of topological degree theory, critical point theory, partial order theory, local and global bifurcation theory, etc.Many mathematicians have made significant contributions to nonlinear functional analysis, e.g. L. Lusternick, L. Schnirelman, M. Morse, R. S. Palais, S. Smale, E. Rothe, M. A. Krasonsel'skii, H. Amann, A. Ambrosetti, P. Rabinowitz, I. Ekeland, H. Brezis, L. Nirenberg. Many well known Chinese mathematicians, e.g. Kung-Ching Chang, Dajun Guo, Wenyuan Chen, Jingxian Sun, have also done good works in various fields in nonlinear functional analysis (see [2,3,7,15,16,32,35,41,45,47,48,58,64,73, 78,91,94]).In this thesis, we mainly explore the existence results for a class of Schrodinger-Maxwell equations using variational methods, and the existence results for a class of fractional differential equations by upper and lower solutions method and fixed point theorem respectively. Schrodinger-Maxwell equations is obtained while looking for existence of standing waves for the nonlinear Schrodinger equations interacting with an unknown electrostatic field. For more of its physical background we refer to [29,39] and the references therein. Recently, the existence results of the standing wave solutions for the Schrodinger-Maxwell equations have been widely studied by many researchers, see e.g. [17,19,36-39,60,82,93,99,100]. Differential equations of fractional order occur more frequently in different research areas and engineering, such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, control of dynamical systems, etc. Recently, many researchers paid attention to existence result of solution of the initial value problem and boundary value problem for fractional differential equations, such as [13,42,43,57,61,63,75,80,96,98]. Some recent contributions to the theory of fractional differential equations can be seen in [10,44,62,70,72,74,76,85].In Chapter 1, we first introduce the history, basic idea and recent progress of variational methods. Then in section 2, we introduce Sobolev space and correspond-ing embedding theorems. In section 3, we present the definition of weak solution of a boundary value problem for elliptic PDE, (P.S.) condition and several minimax theo-rems. In section 4, we recall some important inequalities and several basic theorems of Lebesgue integral theory. In the last section, we introduction the definitions of Riemann-Liouville fractional integral and derivative and their several properties.In Chapter 2, we study the nonlinear stationary Schrodinger-Maxwell equations where for the potential V, we assume V∈C(R3,R), infx∈R3 V(x)≥a1>0, a1>0 and for any M>0, meas{x∈R3|V(x)≤M}<∞. For the nonlinear term f, we don't assume the so called "Ambrosetti-Rabinowitz" type condition, so the verification of (P.S.) condition becomes complicated. In order to overcome this difficulty, we use the variant fountain theorem introduced by Zou [W. Zou Variant fountain theorems and their applications, Manu. Math.104 (2001) 343-358] to get infinitely many large solutions for system (0.1).In Chapter 3, we consider the following initial value problem for fractional differ-ential equation involving Riemann-Liouville sequential fractional derivative where 0< T<+∞and f∈C([0, T]×R×R). By discuss the properties of the well known Mittag-Leffler function we get a comparison result which is important to obtain the main result. Under the assumptions of the existence of upper and lower solutions of problem (0.2) and suitable assumptions on f, we get the existence and uniqueness result of problem (0.2) by using monotone iterative method.In Chapter 4, we consider initial value problem of the following nonlinear sequential fractional differential equation where 0<α≤1,f:[0, 1]×R×R→R is continuous. We make use of the Leray-Schauder type fixed point theorem and Banach contraction principle to get the exis-tence and uniqueness of solutions for problem (0.3). Unlike the monotone iterative method in the previous chapter, we don't need to assume the existence of a pair of upper and lower solutions for problem (0.3).