Applications Of Critical Point Theory To P-Laplacian Difference Equations

In this paper,using critical point theory such as mountain pass lemma,linking the-orem,Clark theorem and so on,we study the existence of periodic solutions of a class difference equations with p-Laplacian operator and the existence of solutions of a class difference equation with p-Laplacian operator under periodic boundary conditions.The first chapter presents the problem's background and state the main results of this thesis.We also list some preliminary knowledge which is needed later.In the second chapter,we consider the existence of periodic solutions of following p-Laplacian difference equation-△[φp(△χ(k-1))]+μa(k)φp(χ(k))=f(k,χ(k)),k∈Z.Where△χ(k)=χ(k+1)-χ(k),p>1 is a constant,φp(s)=[s]p-2S,the continuous function f is T-periodic about the first variable,{a(k)}is a positive sequence with period T. By establishing the suitable variational functional,we transferred the existence of periodic solution of the above problem to the existence of the functional in certain space. First we discussed the existence of one solutions and two nontrivial periodic solutions for the above problem whenμ=0.Then the existence criteria of a positive solution and a negative nontrivial periodic solution for the problem whenμ=1 are established.The main tools are direct method,mountain pass lemma and linking theorem.The third chapter is concerned with the existence of solutions of the following p-Laplacian difference equation with parameter A under periodic boundary conditions-△[φp(△χ(k-1))]+q(k)φp(χ(k))=λf(k,χ(k)),k∈[1,T].χ(0)=χ(T+1),△χ(0)=△χ(T).Where T≥2 is an integer and[1,T]represents{1, 2,…,T),△χ(k)=χ(k+1)-χ(k),p>1 is a constant,{q(k)}is a positive sequence.By established the equivalence variational functional and applying mountain pass lemma,Ricceri theorem and Clark theorem,We obtained the existence of two positive solutions,three solutions and multiple solutions for the boundary value problem when the parameterλbelongs to some suitable intervals. Finally, we give some examples to illustrate the results.