Existence Of Nontrivial Solutions To Second-order Elliptic Difference Equations

In this thesis, the critical point theory is applied to some Dirichlet boundary valueproblems of second order discrete elliptic equations with a positive parameter. Basedon Linking theorem、Clark theorem and Morse theory, respectively, some existenceand multiplicity results of nontrivial solutions are established.In Chapter1, we introduce the historical background and the recent developmentof boundary value problems for elliptic difference equations. We also present the ba-sic knowledge about partial difference equations and the main tools including Linkingtheorem, Clark theorem and Morse theorem, which will be used in the following ar-guments. In addition, the variational structure for discrete elliptic Dirichlet boundaryvalue problems is given.In Chapter2, Linking theorem is applied to discrete elliptic Dirichlet boundaryvalue problems,an existence of at least two nontrivial solutions is established.In chapter3, we establish the existence of at least m pairs nontrivial solutions byClark theorem.In chapter4,an existence of at least two nontrivial solutions is also proved byMorse theory.In each chapter, one example is given to illustrate our main results.