Electron. J. Diff. Eqns., Vol. 2006(2006), No. 131, pp. 1-15.

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

Tsung-fang Wu

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:
 -\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.

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Tsung-Fang Wu
Department of Applied Mathematics
National University of Kaohsiung
Kaohsiung 811, Taiwan
email: tfwu@nuk.edu.tw

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