Existence Of Positive Solutions And Multiple Solutions For Asymptotically Linear Dirichlet Problem

In this paper,we prove the existence of positive solutions and multiple solutions in Sobolev space with variational methods. One hand, our results improve the corresponding results in [6.10.11].On the other hand, some new results as to the existence of multiple solutions are obtained by using several multiplicity theorems in critical points theory.One of the main character of this paper is to present a new way of how to use a mountain pass theorem to prove the existence as to the Dirichlet problem, without assuming Conditions (AR), the other is that we have made great improvements as to the important condition of [6], in other words, we may delete the condition on whichf(x,t)/t is nbndecreasing with respect to t≥0,a.e.x∈Ω. And not only we have used a new technique to verify condition and obtained some more generally results as to the existence of positive solutions, but also we generalized this new idea to asymptotically linear Dirichlet problem for the p-laplacian. As whole, we mainly touch on some aspects as follow:1. Existence of positive solutions for asymptotically linear Dirichlet problem.2. Existence of multiple solutions for asymptotically linear Ddirichlet problem.