Existence And Multiplicity Of Solutions For Nonlinear Difference Double Resonance Equations

In this paper, we mainly study the existence of solutions for the following two discrete boundary conditions equations by using variational method, Morse theory and computations of the critical groups. That is the existence and multiplicity of nontrivial solutions for discrete generalized Emden-Fowler equations with boundary conditions which is resonant at the same eigenvalue both at zero and infinity, and the existence of nontrivial solutions to a discrete Kirchhoff type problem which is resonant both at zero and infinity.The structure of this paper as follows:In the first chapter, we introduce the problem’s background, and research advance of Emden-Fowler equations and Kirchhoff type problems, moreover, state briefly the main results of this thesis.In the second chapter, some necessary preliminaries of the critical groups and Morse theory are stated.In the third chapter, firstly, we give the variational framework of problem (Pi), and analyze the eigenvalue conditions of eigenvalue problem corresponding to problem (Pi). Finally, we give the proofs of the existence and multiplicity of nontrivial solutions for problem (P1).In the fourth chapter, after giving the variational framework of problem (P2), we discuss the eigenvalue conditions of eigenvalue problem corresponding to problem (P3). Furthermore, we give the proofs of the existence of nontrivial solutions for problem (P1) combining with Langrage multplier rule.