We study the regularity of stable solutions to the problem {(−Δ)su=f(u)inB1,u≡0inRn∖B1, where
s∈(0,1). Our main result establishes an L∞ bound for stable and
radially decreasing Hs solutions to this problem in dimensions 2≤n<2(s+2+2(s+1)). In particular, this estimate holds for all s∈(0,1)
in dimensions 2≤n≤6. It applies to all nonlinearities f∈C2.
For such parameters s and n, our result leads to the regularity of the
extremal solution when f is replaced by λf with λ>0. This
is a widely studied question for s=1, which is still largely open in the
nonradial case both for s=1 and s<1