Electron. J. Diff. Eqns., Vol. 2007(2007), No. 69, pp. 1-9.

A fibering map approach to a semilinear elliptic boundary value problem

Kenneth J. Brown, Tsung-Fang Wu

Abstract:
We prove the existence of at least two positive solutions for the semilinear elliptic boundary-value problem
$$ -\Delta u(x) = \lambda a(x) u^q + b(x) u^p \quad\hbox{for } x \in \Omega; \quad u(x) = 0 \quad \hbox{for } x \in \partial \Omega $$
on a bounded region $\Omega$ by using the Nehari manifold and the fibering maps associated with the Euler functional for the problem. We show how knowledge of the fibering maps for the problem leads to very easy existence proofs.

Submitted February 27, 2007. Published May 10, 2007.
Math Subject Classifications: 35J20, 36J65.
Key Words: Semilinear elliptic boundary value problem; variational methods; Nehari manifold; fibering map.

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Kenneth J. Brown
School of Mathematical and Computer Sciences
and the Maxwell Institute, Heriot-Watt University
Riccarton, Edinburgh EH14 4AS, UK
email: K.J.Brown@hw.ac.uk
Tsung-Fang Wu
Department of Applied Mathematics
National University of Kaohsiung
Kaohsiung 811, Taiwan
email: tfwu@nuk.edu.tw

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