An example of self-adjoint and positive operator

Question

Let $\Omega\subset R^2 $ and let $H=H_0^1(\Omega)\cap H^2(\Omega)$ with inner product : $\langle u,v\rangle_H =\langle u,v\rangle_{L^2(\Omega)} + \langle\Delta u,\Delta v\rangle_{L^2(\Omega)}$.

I am looking for an example of a function $f\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega)$ with support $\omega\subset \Omega$ such that the operator $B : y \to f y$, from $H$ to $H$, is such that $B=B^*\ge 0.$

Answer 1

What about $\Omega = \omega = \mathbb R^2$ and $f \equiv 42$?