Are there non-negative functions $f$ satisfying $(-\Delta)^{-1}f \leq f$?

Question

Let $-\Delta$ be the Dirichlet Laplacian on a smooth and bounded domain $\Omega$. Are there any examples of functions $f \in L^2(\Omega)$, $f \geq 0$ not identically zero, such that

$$u \leq f$$ where $u$ is defined as $-\Delta u = f$?

Is there a name for such functions?

Answer 1

With $\Omega=(0,1)$, $f\equiv 2$ and $u(x)=-x^2$ for all $x\in\Omega$, you get $$ -\Delta u = 2 = f \quad \text{and} \quad u \le 0 < 2 = f \quad \text{in $\Omega$.}$$ So yes, functions with this property exist. But I don't know whether there is a name for them or whether they are particularly interesting.