A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations

Abstract

We study stable solutions to the equation $(-\Delta {)}^{1/2}u=f(u)$, posed in a bounded domain of ${\mathbb{R}}^n$. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions $n\le 4$. 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 $n\le 4$, a universal $H^{1/2}$ estimate in all dimensions. Our $L^{\infty }$ bound is expected to hold for $n\le 8$, but this has been settled only in the radial case or when $f(u)=\lambda e^u$. For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when $f(u)=\lambda e^u$, even in the radial case.

38 pages, 1 figure. Version 2 contains a new Liouville result (Corollary 1.5) which is a corollary of our main estimate. Version 3 contains a short comment (Remark 3.1) on the difficulties of extending the result to other powers of the Laplacian. Version 4 is the final version after referee report and proofs, we have added a comment to facilitate the interpretation of the colors of Figure 1

References in corpus (1)

• A Liouville type theorem for fractional elliptic equation with exponential nonlinearity

Cited by in corpus (2)

• Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results
• Stable solution and extremal solution for fractional $p$-Laplacian