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The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function英文.pdf 19页

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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 Abstract 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 1. Introduction 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: N 2 Corresponding author. Fax: 131-451-3249. E-mail address: k.j.brown@ma.hw.ac.uk (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. doi:10.1016/S0022-0396(03)00121-9

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