Qualitative Study For Several Classes Of Partial Difference Equations

This dissertation mainly studies the qualitative properties of several classes of partial difference equations,including boundary value problems,periodic solutions and ground state solutions.By establishing corresponding basic function spaces and constructing appropriate variational structures,seeking solutions of problems is transformed into looking for critical points for variational functionals.The existence and multiplicity of the solutions for considered problems are obtained by applying various methods and techniques in critical point theory,which improve and supplement the relevant results of existing literature.In addition,we give some examples to illustrate our main results.The dissertation is divided into five chapters and organized as follows:Chapter 1 introduces the historical background of the topic,research progress,our main work and some basic knowledge involved in the dissertation.Chapter 2 considers the Dirichlet boundary value problems of the partial difference equations with -Laplacian operator and mean curvature operator,respectively.Using a three critical points theorem,we give the ranges of parameters to ensure that the considered problems possess at least three solutions.Furthermore,based on strong maximum principles,we obtain the conditions that the problems possess at least two positive solutions and at least three positive solutions.Chapter 3 discusses a second-order partial difference equation.By means of Linking theorem and Saddle point theorem,we give a series of sufficient conditions on the existence of periodic solutions for the equation when the nonlinearity 1)is superlinear,sublinear and asymptotically linear,respectively.Some conditions are weaker than the corresponding conditions on the existence of periodic solutions for the difference equation.And,we establish a necessary and sufficient condition on the existence of a unique periodic solution for the equation when 1)grows linearly.Moreover,we also get the nonexistence of nontrivial periodic solutions for the equation.A partial difference equation with periodic coefficients is studied in Chapter4.When the nonlinearity allows for changing sign and satisfies mixed superquadratic and asymptotically quadratic at the origin and infinity,we prove that the equation exists a ground state solution of Nehari-Pankov type by utilizing non-Nehari manifold method and the generalized linking theorem.Chapter 5 is the summary of this dissertation and the prospect for future research.