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Constrained critical points and eigenvalue approximation for semilinear elliptic operators

Constrained critical points and eigenvalue approximation for semilinear elliptic operators Abstract. Let be a bounded open set in R N N b 2 and let L 0 be a uniformly elliptic, for1 mally selfadjoint second order operator acting in H0 . Given an eigenvalue m0 of L 0 , we study its stability under addition to L 0 of a nonlinear term of the form mxY s, under various di¨erent conditions on the function m X Â R 3 R. Bounds on the perturbed eigenvalue mr (associated with eigenfunctions u with kmkL2 r) are established, and the connection with bifurcation from the trivial solutions is discussed. The main tool is a Constrained Saddle Point Theorem for functionals on spherelike submanifolds of a Banach space. 1991 Mathematics Subject Classi®cation: 35P30, 58E05, 58E07, 35J65, 47H12. 1. Introduction Let be a bounded open set in R N N b 2 with smooth boundary q, and let 1X0 N q qu L0 u À aij x a0 xu qxi qxj iY j1 be a uniformly elliptic, formally selfadjoint operator with bounded measurable coe½cients a0 and aiY j ajY i iY j 1Y F F F Y N. As is well known, the eigenvalue problem for L0 in subject to e.g. Dirichlet boundary http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Constrained critical points and eigenvalue approximation for semilinear elliptic operators

Forum Mathematicum , Volume 11 (4) – Jun 1, 1999

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Publisher
de Gruyter
Copyright
Copyright (c)1999 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1999.009
Publisher site
See Article on Publisher Site

Abstract

Abstract. Let be a bounded open set in R N N b 2 and let L 0 be a uniformly elliptic, for1 mally selfadjoint second order operator acting in H0 . Given an eigenvalue m0 of L 0 , we study its stability under addition to L 0 of a nonlinear term of the form mxY s, under various di¨erent conditions on the function m X Â R 3 R. Bounds on the perturbed eigenvalue mr (associated with eigenfunctions u with kmkL2 r) are established, and the connection with bifurcation from the trivial solutions is discussed. The main tool is a Constrained Saddle Point Theorem for functionals on spherelike submanifolds of a Banach space. 1991 Mathematics Subject Classi®cation: 35P30, 58E05, 58E07, 35J65, 47H12. 1. Introduction Let be a bounded open set in R N N b 2 with smooth boundary q, and let 1X0 N q qu L0 u À aij x a0 xu qxi qxj iY j1 be a uniformly elliptic, formally selfadjoint operator with bounded measurable coe½cients a0 and aiY j ajY i iY j 1Y F F F Y N. As is well known, the eigenvalue problem for L0 in subject to e.g. Dirichlet boundary

Journal

Forum Mathematicumde Gruyter

Published: Jun 1, 1999

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