Positive Solutions Of A Sublinear Elliptic Neumann Problem

In this paper,we consider the existence and stability of positive solutions for the following problem(pa,q)where ? is a bounded smooth domain in RN(V?1),0<q<1 is a real parameter,a(x)?Lr(?),r>N.Firstly,under the appropriate conditions,we can get the existence of a positive solution for the equation(pa,q)by the sub-super solution method.Then we use the Lyapunov-Schmidt reduction process divergence technique to prove the existence of q0=q0(a)>0,and prove that the equation(pa,q)has positive solutions when q0<q<1.This article also proves the asymptotic behavior and stability of the positive solution uq when q?1-,.In addition,when ? is the ball and a is a radial function,we give some conditions for a and q to ensure the existence of positive solutions.