The unique continuation property of sublinear equations

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 $-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{\sigma -1}u$ in a domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, 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 $\mathrm{\Omega }$, then it vanishes necessarily in the whole $\mathrm{\Omega }$. We then extend the result to equations with variable coefficients operators and inhomogeneous right-hand side.