ARTICLE IN PRESS
J. Differential Equations 193 (2003) 481–499
The Nehari manifold for a semilinear elliptic
equation with a sign-changing weight function
K.J. Brown and Yanping Zhang1
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 7HQ, UK
Received September 27, 2002
The Nehari manifold for the equation DuexT ? laexTuexT t bexTjuexTjn1uexT for x AO
together with Dirichlet boundary conditions is investigated. Exploiting the relationshi
between the Nehari manifold and ?brering maps (i.e., maps of the form t -J etuT where J is the
Euler functional associated with the equation) we discuss how the Nehari manifold changes as
l changes and show how existence and non-existence results for positive solutions of the
equation are linked to properties of the manifold.
r 2003 Elsevier Science (USA). All rights reserved.
Keywords: Variational methods; Nehari manifold; Inde?nite weight functions
Consider the semilinear boundary value problem
DuexT ? laexTuexT t bexTjuexTjn1uexT for x AO; e1:1T
uexT ? 0 for x A@O; e1:2T
where O is a bounded region with smooth boundary in RN ; l 40 is a real parameter,
N t2 % %
1on o and a; b : O -R are smooth functions which change sign in O:
Corresponding author. Fax: 131-451-3249.
E-mail address: firstname.lastname@example.org (K.J. Brown).
1 Su orted by a James Watt Scholarshi pfrom Heriot-Watt University.
0022-0396/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.