Nonlinear Discrete Boundary Value Problems With ?_c-Laplacian

Based on the critical point theory,we discuss the existence and multiplicity of nontrivial solutions for nonlinear discrete Dirichlet boundary value problems with?_c-Laplacian in this paper.In the basic function space,we establish a variational framework for boundary value problems.By using the three critical points theorem and two nontrivial critical points theorem of G.Bonanno,we discuss the existence of critical points for variational functionals.Thus,we obtain the existence of nontrivial solutions for the boundary value problem.Furthermore,we obtain sufficient conditions for the existence of positive solutions of the boundary value problems by using the strong maximum principle,This paper is divided into three chapters,and the corresponding contents of each chapter are as follows:In the chapter 1,we briefly describe the background,research significance and research progress of the issues discussed in this paper.At the same time,for the convenience,we also list the basic knowledge needed in this paper.In the second chapter,we investigate the existence and multiplicity of nontrivial solutions for discrete boundary value problems involving?_c-Laplacian.When the nonlinear function satisfies some conditions,we obtain sufficient conditions for the existence of at least two nontrivial solutions and at least three nontrivial solutions respectively.Besides,we also deduce the existence of positive solutions from the Maximum Principle.In chapter three,we consider a more general nonlinear boundary value problem than in chapter 2.When the nonlinear term satisfies the appropriate conditions and the parameters lie in a suitable interval,we obtain several sufficient conditions for the existence of nontrivial solutions.