Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains

Takahiro Hashimoto, Mitsuharu Otani

    Research output: Contribution to journalArticle

    8 Citations (Scopus)

    Abstract

    The nonexistence of nontrivial solution is discussed for the following quasilinear elliptic equations: (equation presented) for the case where Ω is an exterior domain such that Ω = RN0, with Ω0 bounded and starshaped. It should be noted that because of the degeneracy of the equation at ∇u = 0, the nontrivial solutions of (E) are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When Ω is bounded, the second author introduced a "Pohozaev-type inequality" valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.

    Original languageEnglish
    Pages (from-to)267-290
    Number of pages24
    JournalHouston Journal of Mathematics
    Volume23
    Issue number2
    Publication statusPublished - 1997

    Fingerprint

    Nonlinear Elliptic Equations
    Exterior Domain
    Nonexistence
    Weak Solution
    Nontrivial Solution
    Exterior Problem
    Valid Inequalities
    Quasilinear Elliptic Equation
    Degeneracy
    Differentiable
    Duality
    Interior

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

    Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains. / Hashimoto, Takahiro; Otani, Mitsuharu.

    In: Houston Journal of Mathematics, Vol. 23, No. 2, 1997, p. 267-290.

    Research output: Contribution to journalArticle

    @article{4c9785c42e414144a47d8c05a4d171f1,
    title = "Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains",
    abstract = "The nonexistence of nontrivial solution is discussed for the following quasilinear elliptic equations: (equation presented) for the case where Ω is an exterior domain such that Ω = RN \Ω0, with Ω0 bounded and starshaped. It should be noted that because of the degeneracy of the equation at ∇u = 0, the nontrivial solutions of (E) are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When Ω is bounded, the second author introduced a {"}Pohozaev-type inequality{"} valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.",
    author = "Takahiro Hashimoto and Mitsuharu Otani",
    year = "1997",
    language = "English",
    volume = "23",
    pages = "267--290",
    journal = "Houston Journal of Mathematics",
    issn = "0362-1588",
    publisher = "University of Houston",
    number = "2",

    }

    TY - JOUR

    T1 - Nonexistence of weak solutions of nonlinear elliptic equations in exterior domains

    AU - Hashimoto, Takahiro

    AU - Otani, Mitsuharu

    PY - 1997

    Y1 - 1997

    N2 - The nonexistence of nontrivial solution is discussed for the following quasilinear elliptic equations: (equation presented) for the case where Ω is an exterior domain such that Ω = RN \Ω0, with Ω0 bounded and starshaped. It should be noted that because of the degeneracy of the equation at ∇u = 0, the nontrivial solutions of (E) are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When Ω is bounded, the second author introduced a "Pohozaev-type inequality" valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.

    AB - The nonexistence of nontrivial solution is discussed for the following quasilinear elliptic equations: (equation presented) for the case where Ω is an exterior domain such that Ω = RN \Ω0, with Ω0 bounded and starshaped. It should be noted that because of the degeneracy of the equation at ∇u = 0, the nontrivial solutions of (E) are not twice differentiable. Therefore there arises the necessity of discussing the nonexistence of solutions in a framework of weak solutions. When Ω is bounded, the second author introduced a "Pohozaev-type inequality" valid for a class of weak solutions, which is effective enough for discussing the nonexistence. We here give an exterior-domain version of Pohozaev-type inequality, whence some results on the nonexistence of solutions are derived. These suggest that concerning the existence and nonexistence of nontrivial solutions, there may exist an interesting duality between the interior problems and the exterior problems for starshaped domains.

    UR - http://www.scopus.com/inward/record.url?scp=0031475170&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0031475170&partnerID=8YFLogxK

    M3 - Article

    AN - SCOPUS:0031475170

    VL - 23

    SP - 267

    EP - 290

    JO - Houston Journal of Mathematics

    JF - Houston Journal of Mathematics

    SN - 0362-1588

    IS - 2

    ER -