The extremal solution for the fractional Laplacian

TL;DR

This paper extends known results about extremal solutions of the Laplacian to the fractional Laplacian, establishing boundedness conditions in various dimensions and nonlinearities, and providing new regularity estimates.

Contribution

It generalizes extremal solution properties to the fractional Laplacian, including boundedness and regularity results, and develops new $L^q$ and $C^eta$ estimates for solutions.

Findings

Extremal solutions are bounded for $n<4s$ with convex nonlinearities.

For exponential and power nonlinearities, boundedness holds when $n<10s$.

The extremal solution is in $H^s( ^n)$ for convex domains.

Abstract

We study the extremal solution for the problem $(-\Delta)^s u=\lambda f(u)$ in $\Omega$, $u\equiv0$ in $\R^n\setminus\Omega$, where $\lambda>0$ is a parameter and $s\in(0,1)$. We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions $n<4s$. We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever $n<10s$. In the limit $s\uparrow1$, $n<10$ is optimal. In addition, we show that the extremal solution is $H^s(\R^n)$ in any dimension whenever the domain is convex. To obtain some of these results we need $L^q$ estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with $L^p$ data. We prove optimal $L^q$ and $C^\beta$ estimates, depending on the value of $p$. These estimates follow from classical embedding results for the Riesz potential in $\R^n$. Finally, to prove the $H^s$ regularity of the extremal solution we need an $L^\infty$ estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.