Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions u(x) >= 0 to the semilinear diffusion equation Lu = f (u), posed in a bounded domain Omega subset of R-N with appropriate homogeneous Dirichlet or outer boundary conditions. The operator L may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian (-Delta)(s) (0 < s < 1) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity f is increasing and looks like a power function f (u) similar to u(p),with p <= 1. The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Holder continuous and even classical (when the operator allows for it). In addition, we get Holder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number 2s/1 p goes below the exponent gamma is an element of (0, 1] corresponding to the Holder regularity of the first eigenfunction L Phi(1) = lambda(1) Phi(1). Indeed a change of boundary regularity happens in the different regimes 2s/1-p (>)=(<) gamma, and in particular a logarithmic correction appears in the critical case 2s/1-p = gamma. For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range 0 < s <= (1 - p)/2.