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(Super)Critical nonlocal equations with periodic boundary conditions

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Abstract

In this paper, we discuss the existence and multiplicity of periodic solutions for a class of parametric nonlocal equations with critical and supercritical growth. It is well known that these equations can be realized as local degenerate elliptic problems in a half-cylinder of \(\mathbb {R}^{N+1}_{+}\) together with a nonlinear Neumann boundary condition, through the extension technique in periodic setting. Exploiting this fact, and by combining the Moser iteration scheme in the nonlocal framework with an abstract multiplicity result valid for differentiable functionals due to Ricceri, we show that the problem under consideration admits at least three periodic solutions with the property that their Sobolev norms are bounded by a suitable constant. Finally, we provide a concrete estimate of the range of these parameters by using some properties of the fractional calculus on a specific family of test functions. This estimate turns out to be deeply related to the geometry of the domain.

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Authors and Affiliations

  1. Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino ‘Carlo Bo’ Piazza della Repubblica, 13 61029, Urbino, Pesaro e Urbino, Italy

    Vincenzo Ambrosio

  2. Institut de Recherche en Mathématique et Physique Université Catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    Jean Mawhin

  3. Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di Vito, 89100, Reggio Calabria, Italy

    Giovanni Molica Bisci

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  1. Vincenzo Ambrosio
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  2. Jean Mawhin
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  3. Giovanni Molica Bisci
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Correspondence to Jean Mawhin.

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Ambrosio, V., Mawhin, J. & Bisci, G.M. (Super)Critical nonlocal equations with periodic boundary conditions. Sel. Math. New Ser. 24, 3723–3751 (2018). https://doi.org/10.1007/s00029-018-0398-y

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