A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations
Entity
UAM. Departamento de MatemáticasPublisher
ElsevierDate
2022-02-11Citation
10.1016/j.jde.2022.02.001
Journal of Differential Equations 317 (2002): 153-195
ISSN
0022-0396 (print); 1090-2732 (online)DOI
10.1016/j.jde.2022.02.001Funded by
Both authors are supported by grants MTM2017-84214-C2-1-P and RED2018-102650-T funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. They are members of the Barcelona Graduate School of Mathematics (BGSMath) and of the Catalan research group 2017 SGR 01392. The second author acknowledges fnancial support from the Spanish Ministry of Economy and Competitiveness (MINECO), through the Mar´ıa de Maeztu Program for Units of Excellence in R&D MDM-2014-0445, as well as from the EPSRC grant EP/S03157X/1Project
Gobierno de España. MTM2017-84214-C2-1-P; Gobierno de España. MDM-2014-0445; Gobierno de España. RED2018-102650-TEditor's Version
https://doi.org/10.1016/j.jde.2022.02.001Subjects
Half-Laplacian; Stable Solutions; Extremal Solution; Interior Estimates; Dirichlet Problema; MatemáticasRights
© 2022 Elsevier Inc.Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
Abstract
We study stable solutions to the equation (−Δ)1/2u=f(u), posed in a bounded domain of Rn. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions n≤4. This result, which was known only for n=1, follows from a new interior Hölder estimate that is completely independent of the nonlinearity f. A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobolev estimates that allow us to extend the techniques developed by Cabré, Figalli, Ros-Oton, and Serra for the Laplacian. In this way we obtain, besides the Hölder bound for n≤4, a universal H1/2 estimate in all dimensions. Our L∞ bound is expected to hold for n≤8, but this has been settled only in the radial case or when f(u)=λeu. For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when f(u)=λeu, even in the radial case
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Google Scholar:Cabré, Xavier
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Sanz-Perela, Tomás
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