A positive solution for an asymptotically linear elliptic problem on ℝN autonomous at infinity

Louis Jeanjean, Kazunaga Tanaka

    Research output: Contribution to journalArticle

    90 Citations (Scopus)

    Abstract

    In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on ℝN. 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.

    Original languageEnglish
    Pages (from-to)597-614
    Number of pages18
    JournalESAIM Control, Optimisation and Calculus of Variations
    Issue number7
    Publication statusPublished - 2002

    Fingerprint

    Asymptotically Linear
    Regain
    Elliptic Problems
    Compactness
    Positive Solution
    Infinity
    Concentration-compactness
    A Priori Bounds

    Keywords

    • Asymptotically linear problems in ℝ
    • Elliptic equations
    • Lack of compactness

    ASJC Scopus subject areas

    • Control and Systems Engineering

    Cite this

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    AB - In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on ℝN. 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.

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