Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 131, pp. 1-15.

Title: A semilinear elliptic problem involving nonlinear boundary 
       condition and sign-changing potential

Author: Tsung-fang Wu (National Univ. of Kaohsiung, Taiwan)

Abstract: 
 In this paper, we study the multiplicity of nontrivial nonnegative
 solutions for a semilinear elliptic equation involving nonlinear
 boundary condition and sign-changing potential. With the help of
 the Nehari manifold, we prove that the semilinear elliptic
 equation:
 $$\displaylines{
 -\Delta u+u=\lambda f(x)|u|^{q-2}u \quad \hbox{in }\Omega , \cr
 \frac{\partial u}{\partial \nu }=g(x)|u|
 ^{p-2}u \quad \hbox{on }\partial \Omega ,
 }$$
has at least two nontrivial nonnegative solutions for $\lambda $
is sufficiently small.

Submitted July 6, 2006. Published October 17, 2006.
Math Subject Classifications: 35J65, 35J50, 35J55.
Key Words: Semilinear elliptic equations; Nehari manifold; 
           Nonlinear boundary condition.