Assume that $\Omega\subset\mathbb{R}^{n}$ is a connected open set and $u\in W^{1,2}_{loc}(\Omega)$ is a weak solution to the equation

$$\partial_{i}(a^{ij}(x)u_{j})+b^{i}(x)u_{i}+c(x)u=0,$$

where $a^{ij}(x)$ is Lipschitz and uniformly elliptic and $b^{i}(x),c(x)$ are bounded measurable. If $u=0$ on some open subset of $\Omega$, then $u\equiv 0$ in $\Omega$.