Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian

Tomás Sanz-Perela

doi:10.3934/cpaa.2018121

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2018, Volume 17, Issue 6: 2547-2575. Doi: 10.3934/cpaa.2018121

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Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian

Tomás Sanz-Perela

Universitat Politècnica de Catalunya and BGSMath, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain

Received: December 31, 2017

Revised: February 28, 2018

Published: June 2018

The author is supported by MINECO grants MDM-2014-0445 and MTM2014-52402-C3-1-P. He is member of the Barcelona Graduate School of Mathematics and part of the Catalan research group 2014 SGR 1083.

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Abstract

We study the regularity of stable solutions to the problem

$\begin{align}\left\{ \begin{gathered} {\left( { - \Delta } \right)^s}&u = f\left( u \right)&{\text{in}}\;\;{B_1}, \hfill \\ &u \equiv 0&{\text{in}}\;\;{{\mathbb{R}}^n}\backslash {B_1}, \hfill \\ \end{gathered} \right.\end{align}$

where $s∈(0,1)$ . Our main result establishes an $L^∞$ bound for stable and radially decreasing $H^s$ solutions to this problem in dimensions $2 ≤ n < 2(s+2+\sqrt{2(s+1)})$ . In particular, this estimate holds for all $s∈(0,1)$ in dimensions $2 ≤ n≤ 6$ . It applies to all nonlinearities $f∈ C^2$ .

For such parameters $s$ and $n$ , our result leads to the regularity of the extremal solution when $f$ is replaced by $λ f$ with $λ > 0$ . This is a widely studied question for $s = 1$ , which is still largely open in the nonradial case both for $s = 1$ and $s < 1$ .

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Mathematics Subject Classification: Primary: 35J61, 35R11; Secondary: 35B45, 35B35, 35B65, 35J70.

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