Applications Of Morse Theory To Dirichlet Boundary Value Problems Of Discrete Elliptic Equations

This paper mainly discusses the existence and multiplicity of nontrivial solutions for two types of Dirichlet boundary value problems of partial difference equations.By variational methods,we construct a suitable function space and transform the boundary value problems of the partial difference equations into the corresponding functional critical point problems.Applying Morse theory,the existence of non-zero critical points for the variational functionals are obtained,and a series of results on the existence and multiplicity of nontrivial solutions for the Dirichlet boundary value problems of the partial difference equations are established.Moreover,examples and numerical simulations are presented to demonstrate the feasibility of the obtained results.The paper are composed of five chapters and arranged as follows:The first chapter introduces the background and significance of the topic,the current research status as well as the main work of this paper.Also,the main tools used in this paper are listed in this chapter.The existence and multiplicity of nontrivial solutions subject to the following discrete elliptic Dirichlet boundary value problem are considered in Chapter 2,3 and 4.In Chapter 2,based on Morse theory together with variational techniques,we achieve that the problem possesses at least three nontrivial solutions,including one positive solution and one negative solution.Further,if the nonlinear term f satisfies asymptotic linearity at 0,we obtain a series of results that the equation admits at least four or two nontrivial solutions.Finally,examples and numerical simulations are provided to verify the correctness of obtained conclusions.If the nonlinear term f satisfies double resonance at infinity and 0,Chapter 3 considers sufficient conditions for the existence of at least five,two,and one nontrivial solutions by verifying the Morse equalities and calculating the critical groups of non-zero critical points.Chapter 4 discusses the problem when the nonlinear term f resonates at infinity.Using the Morse equalities,we acquire that the equation has at least two nontrivial solutions.At same time,five examples are also provided to illustrate the feasibility of the obtained results.In Chapter 5,we focus on studying the existence of nontrivial solutions to self-adjoint partial difference equationΔ1[p(i-1,j)Δ1u(i-1,j)]+Δ2[r(i,j-1)Δ2u(i,j-1)]+q(i,j)u(i,j)=-f((i,j),u(i,j)),(i,j)∈Ωwith Dirichlet boundary conditions u(i,0)=u(i,T2+1)=0,i∈Z(1,T1),u(0,j)=u(T1+1,j)=0,j∈Z(1,T2),where the nonlinear term f resonates at infinity.By Morse theory,cut-off technique and mountain pass lemma,we prove that the equation has at least three nontrivial solutions,one of which is positive and the other is negative.