Existence Of Solutions For Three Classes Of P-Laplacian And P(x)-Laplacian Equations

In recent years,the boundary value problems of p-Laplacian equations and p(x)-Laplaci-an equations have been paid more and more attention by researchers at home and broad.Besides their application in mathematics,they are widely used in the aspects of newtonian mechanics,cosmic physics,phasma problem,elastic mechanics,electrical fluid mechanics and so on.At present,there are many achievements in the study of the boundary value problem for p-Laplacian equations and p(x)-Laplacian equations.In this paper we talks about the existence of solutions for three classes of p-Laplacian and p(x)-Laplacian equations.Firstly,we talks about the existence of solutions when p(x)? p which is the kind of p-Laplacian equations.In this section,we get three existence Theorems from three equa-tions.Our proof combines the presence of sub and supersolution with the pseudomonotone operators theory.These theorems gave a generation to Alves C O's and Covei D P's work.Secondly,we talks about the existence of solutions for three classes of p(x)-Laplacian equations of the boundary problems.In this section we get three existence Theorems from three equations via variational method,Mountain pass lemma and sub and supersolution.It is worth noting that the discussion is based on the theory of the spaces Lp(x)(?)and Wk,p(x)(?).