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.