Let $\Omega$ be a bounded domain of $R^n$ and let $y\in H^2(\Omega)\cap H_0^1(\Omega)$ such that the set $[x\in \Omega/ y(x)\ne 0]$ has non nul measure and $ \; \frac{\Delta y}{y} 1_{\{x\in \Omega/ y(x)\ne 0\}} \in L^\infty(\Omega).$
Does the following inequality holds? $|\Delta y(x)|\le C |y(x)|,\; a.e \; x\in \Omega$ for some $C>0$.
A very nice result (which is rather unknown / unused in my opinion) is the following: For $y \in H^1(\Omega)$, we have $\nabla y = 0$ a.e. in the set $\{ x \in \Omega : y(x) = 0\}$. This result basically follows from $\nabla(\max\{y,0\}) = \nabla y \, 1_{\{x \in \Omega : y(x) > 0\}}$.
Taking the second derivative, we have for $y \in H^2(\Omega)$, that $\Delta y = 0$ a.e. on $\{x \in \Omega : y(x) = 0\}$. Together with your estimate on the complement set, you have $$|\Delta y(x)| \le C \, |y(x)| \quad\text{f.a.a. }x \in \Omega.$$