*Submitted Paper*

**Inserted:** 26 jul 2017

**Last Updated:** 26 jul 2017

**Year:** 2017

**Links:**
Link arXiv

**Abstract:**

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

^{\sigmaâˆ’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.