The Presence Of Several Types Of Nonlinear Partial Differential Equations,

In view of the extensive connection between the Partial differential equation (PDE) and social activity, people have paid more attention to the study of them and got many important results, which made the theory on PDE more perfect. We based on other people's job, proved the existence of nontrivial solutions for some special elliptic equations by variational methods including Mountain Pass Lemma and concentration Compact Principle. In this paper, firstly, we introduced the eigenvalue problem of the p-Laplace equation and mainly talked about some important propositions of the first eigenvalue including isolated, simple and closeness. Secondly, we studied the Dirichlet problem of p-Laplacian:And proved the existence of the positive solution without the (AR) condition which is very useful for the proof of the bounded of sequence.Then, without the compact imbedding, we proved the existence of the nontrivial solution for the problem:When f(x,u) =| u|^p-2u, the author of [6] got the existence of multiple solutions, and we found that there still existed some nonzero solutions associated to the parameterλthough the perturbation has higher (lower than the critical exponent) degree.Finally, under the general format of operator, we proved the existence the nontrivial solution of the following elliptic system:Such operators may be the p-Laplace operator, Laplace operator or curvature operator under some conditions, to some extent, the result was more general.