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A positive solution for an asymptotically linear elliptic problem on$\mathbb{R}^N$ autonomous at infinity

Published online by Cambridge University Press:  15 September 2002

Louis Jeanjean  and
Kazunaga Tanaka
Louis Jeanjean
Affiliation:
Équipe de Mathématiques, UMR 6623 du CNRS, Université de Franche-Comté, 16 route de Gray, 25030 Besançon, France; jeanjean@math.univ-fcomte.fr.
Kazunaga Tanaka
Affiliation:
Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169, Japan; kazunaga@mn.waseda.ac.jp.
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Abstract

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on $\xR^N$. The main difficulties to overcome are the lack of a priori bounds for Palais–Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the “Problem at infinity” is autonomous, in contrast to just periodic, can be used in order to regain compactness.

Keywords

Elliptic equationsasymptotically linear problems in $\xR^N$lack of compactness.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 597 - 614
Copyright
© EDP Sciences, SMAI, 2002

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