A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations
EntityUAM. Departamento de Matemáticas
10.1016/j.jde.2022.02.001Journal of Differential Equations 317 (2002): 153-195
ISSN0022-0396 (print); 1090-2732 (online)
ProjectGobierno de España. MTM2017-84214-C2-1-P; Gobierno de España. PID2019-111338RB-I00; Gobierno de España. MDM-2014-0445
SubjectsHalf-Laplacian; Stable Solutions; Extremal Solution; Interior Estimates; Dirichlet Problema; Matemáticas
Rights© 2022 Elsevier Inc.
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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