Abstract
We consider the following fractional semilinear Neumann problem on a smooth bounded domain \(\Omega \subset \mathbb {R}^n\), \(n\ge 2\),
where \(\varepsilon >0\) and \(1<p<(n+1)/(n-1)\). This is the fractional version of the semilinear Neumann problem studied by Lin–Ni–Takagi in the late 1980’s. The problem arises by considering steady states of the Keller–Segel model with nonlocal chemical concentration diffusion. Using the semigroup language for the extension method and variational techniques, we prove existence of nonconstant smooth solutions for small \(\varepsilon \), which are obtained by minimizing a suitable energy functional. In the case of large \(\varepsilon \) we obtain nonexistence of nonconstant solutions. It is also shown that as \(\varepsilon \rightarrow 0\) the solutions \(u_\varepsilon \) tend to zero in measure on \(\Omega \), while they form spikes in \(\overline{\Omega }\). The regularity estimates of the fractional Neumann Laplacian that we develop here are essential for the analysis. The latter results are of independent interest.
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Acknowledgments
This research was motivated from discussions between the second author and Christian Kuehn. We thank Laurent Saloff-Coste and Jiaping Wang for very interesting discussions regarding the Neumann heat kernel, and to Luis Caffarelli for pleasant conversations about this work. We are also grateful to Benedetta Pellacci for pointing out a computational mistake in an earlier version of this paper, as well as to the referee for very useful detailed remarks that helped us to improve the presentation of the results. The first author is grateful to the Dipartimento di Ingegneria at Università degli Studi di Napoli “Parthenope” for their kind hospitality during several visits. The authors have been partially supported by MTM2011-28149-C02-01 from Spanish Government and the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), Italy.
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Communicated by L. Ambrosio.
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Stinga, P.R., Volzone, B. Fractional semilinear Neumann problems arising from a fractional Keller–Segel model. Calc. Var. 54, 1009–1042 (2015). https://doi.org/10.1007/s00526-014-0815-9
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DOI: https://doi.org/10.1007/s00526-014-0815-9