Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

Abstract

We investigate quantitative properties of nonnegative solutions $u(x)\ge 0$ to the semilinear diffusion equation $\mathcal {L}u= f(u)$, posed in a bounded domain $\Omega \subset \mathbb {R}^N$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $\mathcal {L}$ 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 $f(u)\sim u^p$, with $p\le 1$. 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 Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number $\frac{2s}{1-p}$ goes below the exponent $\gamma \in (0,1]$ corresponding to the Hölder regularity of the first eigenfunction $\mathcal {L}\Phi _1=\lambda _1 \Phi _1$. Indeed a change of boundary regularity happens in the different regimes $\frac{2s}{1-p} \gtreqqless \gamma$, and in particular a logarithmic correction appears in the “critical” case $\frac{2s}{1-p} = \gamma$. For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range $0<s\le (1-p)/2$.