  We derive the unique continuation property of a class of semi-linear elliptic
equations with non-Lipschitz nonlinearities. The simplest type of equations to
which our results apply is given as $-\Delta u = |u|^{\sigma-1} u$ in a domain
$\Omega \subset \mathbb{R}^N$, with $0 \le \sigma <1$. Despite the sublinear
character of the nonlinear term, we prove that if a solution vanishes in an
open subset of $\Omega$, then it vanishes necessarily in the whole $\Omega$. We
then extend the result to equations with variable coefficients operators and
inhomogeneous right-hand side.

Analysis of PDEs