Withdrawal Of The Multiple Linear Elliptic Equations Intends Supercritical Neumann Boundary Value

In the research of quasilinear elliptic equation, quite a number of results .. have appeared on the existence of infinitely many solutions and on the multiplicity of positive solution of Dirichlet problem, but only few results on the existence of infinitely many solutions and the multiplicity of positive solution of Neumann problem. Recently, the remarkable result about Neumann problem is the work of Zhang guiyi and Shen yaotian ON THE EXISTENCE OF INFINITELY MANY SOLUTIONS OF A GENERAL CRITICAL QUASILINEAR ELLIPTIC EQUATION WITH A NONLINEAR CRITICAL BOUNDARY CONDITION.The central work of this paper is to probe into the multiplicity of many positive solutions with a quasi-supercritical boundary condiction. We consider the following problemtherein is the Sobolev critical embeddedN -2ccexponent. We discuss the situation of q,therein, 5 is the N-l dimension Sobolev critical embedded exponent. Due to the extra boundary integration item of the corresponding functional, to make thefunctional hold, we must research it in a new Banach spac. Firstly, we validate the functional J(u) satisfys the given conditions meets the conditions in the Mountain Pass Lemma without (PS) condition, then we use the Mountain Pass Lemma and the famous Ekeland theory to prove the existence of two positive solutions.