Boundary Value Problems Of Difference Equations With Laplacian Operators

In this paper,boundary value problems of several classes of difference equa-tions are studied.By establishing proper variational functional and using critical point theory,the existence and multiplicity of solutions for boundary value prob-lems of difference equations are obtained.The paper is divided into six chapters,the brief outline are as follows:In Chapter 1,we discuss the historical background,recent developments of the researches,research methods and the main work of this paper,we also present some preliminary knowledge needed in this paper.In Chapter 2,we study a class of Dirichlet boundary value problems for second order self-adjoint difference equations with-Laplacian operator.Some sufficient conditions for the existence of three solutions of boundary value problems are ob-tained by means of critical point theory.Furthermore,using the strong maximum principle,we prove that these three solutions are positive.Finally,we give two examples to illustrate our results.In Chapter 3,we study partial discrete Dirichlet boundary value problems with(p,q)-Laplacian operator.By using the critical point theory,we obtain some sufficient conditions for the existence of infinitely solutions of the boundary value problem for the nonlinear term1)which is asymmetric at the origin.To the best of our knowledge,this is the first study of Dirichlet boundary value problem of partial difference equation with perturbed term.Then,an example is given to illustrate our results.Next,by applying critical point theory,we obtain the existence of three solutions of the boundary value problem.Furthermore,we prove that the three solutions are positive by the strong maximum principle.Finally,an example is given to illustrate our results.In Chapter 4,we discuss a class of perturbed partial discrete Dirichlet bound-ary value problems with (?)-Laplacian operator.By using the critical point theory and the strong maximum principle,when the nonlinear term1)satisfies the ap-propriate oscillation condition at the origin,the perturbed term2)satisfies the appropriate hypothesis,and the sign of f((s,t),0) and g((s,t),0)satisfy certain conditions,we obtain some results for the existence of constant sign solutions of boundary value problems with one or two columns converging to zero.An example is given to illustrate our results.In Chapter 5,we consider partial discrete boundary value problems with (?)-Laplacian operator.By using the critical point theory and the strong maximum principle,when the nonlinear term1)satisfies the appropriate oscillation condition at the origin,the perturbed term2)satisfies the appropriate hypothesis,and the sign of f((s,t),0) and g((s,t),0) satisfy certain conditions,we obtain some results for the existence of constant sign solutions of boundary value problems with one or two columns converging to zero.Next,by applying critical point theory,we determine proper intervals of the parametersuch that infinitely many solutions of the boundary value problem are obtained.Furthermore,according to the strong maximum principle established,we give sufficient conditions for the nonlinear term1)to be satisfied to make the boundary value problem possesses infinitely positive solutions.Finally,we give two examples to illustrate our results.In Chapter 6,we summarize this paper and the future research is planned.