Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains

Satoshi Hashimoto, Mitsuharu Otani

    Research output: Contribution to journalArticle

    2 Citations (Scopus)

    Abstract

    The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as -Δu = u β in Ω, ((N + 2)/(N - 2) < β), u = 0 on ∂B, with Ω = ℝN\Ω̄ where Ω0 is the open ball. To recover the compactness of the embedding L β+1(Ω) ⊂ H0 1(Ω, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.

    Original languageEnglish
    Pages (from-to)323-333
    Number of pages11
    JournalDiscrete and Continuous Dynamical Systems
    Volume19
    Issue number2
    Publication statusPublished - 2007 Oct

    Fingerprint

    Nonlinear Elliptic Equations
    Exterior Domain
    Nontrivial Solution
    Elliptic Equations
    Ball
    Nonlinearity
    Singularity
    Order of Growth
    Existence of Positive Solutions
    Symmetric Functions
    Ring or annulus
    Critical Exponents
    Compactness
    Coefficient
    Term
    Class

    Keywords

    • Exterior domain
    • P-Laplacian
    • Supercritical exponent

    ASJC Scopus subject areas

    • Mathematics(all)
    • Discrete Mathematics and Combinatorics
    • Applied Mathematics
    • Analysis

    Cite this

    Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains. / Hashimoto, Satoshi; Otani, Mitsuharu.

    In: Discrete and Continuous Dynamical Systems, Vol. 19, No. 2, 10.2007, p. 323-333.

    Research output: Contribution to journalArticle

    @article{45867f5df66e422390ab12edb71c1a36,
    title = "Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains",
    abstract = "The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as -Δu = u β in Ω, ((N + 2)/(N - 2) < β), u = 0 on ∂B, with Ω = ℝN\Ω̄ where Ω0 is the open ball. To recover the compactness of the embedding L β+1(Ω) ⊂ H0 1(Ω, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.",
    keywords = "Exterior domain, P-Laplacian, Supercritical exponent",
    author = "Satoshi Hashimoto and Mitsuharu Otani",
    year = "2007",
    month = "10",
    language = "English",
    volume = "19",
    pages = "323--333",
    journal = "Discrete and Continuous Dynamical Systems- Series A",
    issn = "1078-0947",
    publisher = "Southwest Missouri State University",
    number = "2",

    }

    TY - JOUR

    T1 - Existence of nontrivial solutions for some elliptic equations with supercritical nonlinearity in exterior domains

    AU - Hashimoto, Satoshi

    AU - Otani, Mitsuharu

    PY - 2007/10

    Y1 - 2007/10

    N2 - The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as -Δu = u β in Ω, ((N + 2)/(N - 2) < β), u = 0 on ∂B, with Ω = ℝN\Ω̄ where Ω0 is the open ball. To recover the compactness of the embedding L β+1(Ω) ⊂ H0 1(Ω, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.

    AB - The existence of positive solutions is discussed for some nonlinear elliptic equations involving the nonlinear terms with the growth order of super-critical exponents in exterior domains of balls such as -Δu = u β in Ω, ((N + 2)/(N - 2) < β), u = 0 on ∂B, with Ω = ℝN\Ω̄ where Ω0 is the open ball. To recover the compactness of the embedding L β+1(Ω) ⊂ H0 1(Ω, we work in the class of radially symmetric functions and introduce a new transformation, which reduces our problems to some nonlinear elliptic equations in annuli but with coefficients which have some singularity on the boundary. The difficulty caused by the singularity on the boundary will be managed by the arguments developed in our previous work.

    KW - Exterior domain

    KW - P-Laplacian

    KW - Supercritical exponent

    UR - http://www.scopus.com/inward/record.url?scp=36248976091&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=36248976091&partnerID=8YFLogxK

    M3 - Article

    AN - SCOPUS:36248976091

    VL - 19

    SP - 323

    EP - 333

    JO - Discrete and Continuous Dynamical Systems- Series A

    JF - Discrete and Continuous Dynamical Systems- Series A

    SN - 1078-0947

    IS - 2

    ER -