The extremal solution for the fractional Laplacian

Abstract

We study the extremal solution for the problem (-¿)su=¿f(u) in O , u=0 in Rn\O , where ¿>0 is a parameter and s¿(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¿1 , n<10 is optimal. In addition, we show that the extremal solution is Hs(Rn) in any dimension whenever the domain is convex. To obtain some of these results we need Lq estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with Lp data. We prove optimal Lq and Cß estimates, depending on the value of p . These estimates follow from classical embedding results for the Riesz potential in Rn . Finally, to prove the Hs regularity of the extremal solution we need an L8 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.