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

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cocvVol7-24.dvi ESAIM: Control, Optimisation and Calculus of Variations August 2002, Vol. 7, 597{614 URL: DOI: 10.1051/cocv:2002068 A POSITIVE SOLUTION FOR AN ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN AUTONOMOUS AT INFINITY � Louis Jeanjean1 and Kazunaga Tanaka2 Abstract. In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on RN . The main di�culties 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 �rst 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 in�nity" is autonomous, in contrast to just periodic, can be used in order to regain compactness. Mathematics Subject Classi�cation. 35J60, 58E05. Received July 9, 2001. Revised January 3, 2002. 1. Introduction In this paper we study the existence of a positive solution u 2 H1(RN ) for an equation of the form −�u+ V (x)u = f(u); x 2 RN ; (1.1) where N � 2 and we assume on the potential V 2 C(RN ;R) (V1) there exists � > 0 such that V (x) � � for all x 2 RN ; (V2) limjxj!1 V (x) = V (1) 2 (0;1) and on the nonlinear term f 2 C(R+;R) (f1) f(s)s−1 ! 0 as s! 0+; (f2) There is a 2]0;1[ such that f(s)s−1 ! a as s! +1 and a > inf �(−� + V (x)); where �(−� + V (x)) denotes the spectrum of the self-adjoint operator −� + V (x) : H2(RN )! L2(RN ): Keywords and phrases: Elliptic equations, asymptotically linear problems in RN , lack of compactness. � The second author is partially supported by Waseda University Grant for Special Research Project 2001A-098. 1 �Equipe de Math�ematiques, UMR 6623 du CNRS, Universit�e de Franche-Comt�e, 16 route de Gray, 25030 Besan�con, France; e-mail: [email protected] 2 Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169, Japan; e-mail: [email protected]

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