A positive solution for an asymptotically linear elliptic problem on autonomous at infinity
Louis Jeanjean1 and Kazunaga Tanaka2
Équipe de Mathématiques, UMR 6623 du CNRS, Université de Franche-Comté,
16 route de Gray, 25030 Besançon, France; firstname.lastname@example.org.
2 Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169, Japan; email@example.com.
Revised: 3 January 2002
In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on . 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.
Mathematics Subject Classification: 35J60 / 58E05
Key words: Elliptic equations / asymptotically linear problems in / lack of compactness.
© EDP Sciences, SMAI, 2002