Boundary Value Problems For Several Classes Of Partial Difference Equations

In this paper,the boundary value problems for several classes of partial difference equations are studied.By establishing proper variational frameworks,we reduce the solutions of boundary value problems to the critical points of corresponding variational functionals.Via using critical point theory,we get the existence of infinitely many solutions and an unbounded sequence of solutions for boundary value problems of partial difference equations.Then the multiplicity of positive solutions of boundary value problems is obtained according to the strong maximum principle.The paper is divided into six chapters,the brief outline are as follows:In Chapter 1,we discuss the historical background of the field,recent developments of the researches and we also state the main work of this paper.At the same time,we present the relevant basic conclusions and some preliminary knowledge that are important for this paper.In Chapter 2,we study the existence of infinitely many solutions for a partial difference Dirichlet boundary value problem with the p-Laplacian.By using critical point theory and the strong maximum principle,we obtain multiple positive solutions of the boundary value problem.And we give two examples to illustrate our results.In Chapter 3,we discuss a general partial difference Neumann boundary value problem involving the p-Laplacian.By constructing the corresponding variational framework and combining critical point theory,we obtain the existence of multiple solutions of the boundary value problem when the parameter ? lies in an proper interval.In Chapter 4,we consider a partial difference Dirichlet boundary value problem with ?c-Laplacian.Via applying critical point theory and the strong maximum principle,we put forward some appropriate conditions for the nonlinear term to be satisfied so that multiple positive solutions and unbounded sequences of positive solutions are obtained.In Chapter 5,we investigate the existence of infinitely many solutions of a class of partial difference Robin boundary value problem with ?p-Laplacian.By applying critical point theory,we determine proper intervals of the parameter ?such that multiple solutions for the boundary value problem are obtained.According to the strong maximum principle,we give sufficient conditions for the nonlinear term to be satisfied to make the boundary value problem admits multiple positive solutions.In Chapter 6,we summarize this paper and plan for future research work.