If $u < 0$ then can I choose $\epsilon$ so that $u+\epsilon\varphi \leq 0$ for $\varphi \in C_c^\infty$?

Question

Here I take $\Omega$ a bounded and smooth domain.

Let $u \in H^1(\Omega)$ satisfy $u < 0$ a.e. Is it true that for every $\varphi \in C_c^\infty(\Omega)$, I can choose $\epsilon$ small enough so that

$$u + \epsilon \varphi \leq 0 \text{ a.e.}$$

is achieved? The problem is the $\epsilon$ should not depend on the $x$, and I am not sure this uniformity holds. Is there anything that I could assume to guarantee it?

Answer 1

This is not true. Let $\Omega$ be the unit ball and consider $u(x) = -\|x\|$.

I think that your requirement is equivalent to $u < -\delta$ on $\Omega$ for some $\delta > 0$.