The Existence Of Multiple Positive Solutions For The Semilinear Elliptic Problem With The Neumann Boundary

In this paper, we consider the existence of multiple positive solutions for the semilinear elliptic problemwhere is a bounded smooth domain with C1 boundary such that ,where 2* = 2N/(N - 2) is the critical exponent for Sobolev embedding, a > 0 is a real parameter, and (x) is some given function in Ca( ) such that in . The function has these characters: it's singular at 0, and it has a critical exponent term for p = 2*- 1 and an inhomogeneous perturbation. Similar equations have been studied by many scholars all over the world. For example, in 1983, the existence of the positive solution for the equation was studied by Brezis and Nirenberg; The existence and nonexistence of the multiple positive solutions for the equation- with Neumann boundary value problem was studiedby Professor Deng Yinbin in 1993; In 1999, the equation was studied by E.Jannelli.In our paper,we obtain a existence result of the minimal positive solution by maximum principle, implicit function theorem. Then we get a non-existence result by virtue of consideration of an eigenvalue problem. At last, we find a point o* > 0 such that the equation has a minimal positive solution if and has no solution if .we set up a new equation, and verify that the corresponding variational functional satisfies the conditions in Mountain Pass Lemma. Then we give a sufficient condition of the existence of critical point. Later, we estimate our variational functional to get a nontrivial solution of the new equation and so the second solution for (*) is obtained.