Behaviour of positive function on compact sets

Question

Let $v$ be any strictly positive bounded function on $\Omega \subset \mathbb{R}^n$ and zero on $\partial \Omega.$

Can we say that for each compact subsets (w.r.t usual topology defined on $\mathbb{R}^n $) of $\Omega$ there exists a constant $c$ such that $v \geq c >0$?

If not, what can be the example?

Answer 1

Let $n=1$, $\Omega =(-1,1)$, $u(\frac 1 n)=\frac 1 n$ for $n=1,2,...$ and $u(x)=1$ for all other $x \in \Omega$, $u(x)=0$ for $x =\pm 1$. Consider the compact set $[0,\frac 1 2]$.

Answer 2

Well let x(n)=p+ (1\n)p with p in the compact set (say a say a closed ball about p lying in your open set and n sufficiently large. Let v(x(n)) =1/n with . v(q)=1 at other points in your open set ( and 0 on the boundary ).