Bifurcation results for a fractional elliptic equation with critical exponent in Rn

Serena Dipierro, María Medina, Ireneo Peral, Enrico Valdinoci

    Research output: Contribution to journalArticle

    24 Citations (Scopus)

    Abstract

    In this paper we study some nonlinear elliptic equations in Rn obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is (-Δ)su=εhu+q+u+pinRn,where s∈ (0 , 1) , n> 4 s, ε> 0 is a small parameter, p=n+2sn-2s, 0 < q< p and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case 0 < q< 1 is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

    Original languageEnglish
    Pages (from-to)183-230
    Number of pages48
    JournalManuscripta Mathematica
    Volume153
    Issue number1-2
    DOIs
    Publication statusPublished - 1 May 2017

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    Elliptic Equations
    Critical Exponents
    Fractional
    Bifurcation
    Elliptic Regularity
    Lyapunov-Schmidt Reduction
    Regularity Theory
    Critical Sobolev Exponent
    Nonlinear Elliptic Equations
    Energy Functional
    Small Parameter
    Regularity
    Perturbation

    Cite this

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    abstract = "In this paper we study some nonlinear elliptic equations in Rn obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is (-Δ)su=εhu+q+u+pinRn,where s∈ (0 , 1) , n> 4 s, ε> 0 is a small parameter, p=n+2sn-2s, 0 < q< p and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case 0 < q< 1 is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.",
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    Bifurcation results for a fractional elliptic equation with critical exponent in Rn. / Dipierro, Serena; Medina, María; Peral, Ireneo; Valdinoci, Enrico.

    In: Manuscripta Mathematica, Vol. 153, No. 1-2, 01.05.2017, p. 183-230.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Bifurcation results for a fractional elliptic equation with critical exponent in Rn

    AU - Dipierro, Serena

    AU - Medina, María

    AU - Peral, Ireneo

    AU - Valdinoci, Enrico

    PY - 2017/5/1

    Y1 - 2017/5/1

    N2 - In this paper we study some nonlinear elliptic equations in Rn obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is (-Δ)su=εhu+q+u+pinRn,where s∈ (0 , 1) , n> 4 s, ε> 0 is a small parameter, p=n+2sn-2s, 0 < q< p and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case 0 < q< 1 is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

    AB - In this paper we study some nonlinear elliptic equations in Rn obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is (-Δ)su=εhu+q+u+pinRn,where s∈ (0 , 1) , n> 4 s, ε> 0 is a small parameter, p=n+2sn-2s, 0 < q< p and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case 0 < q< 1 is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

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