Is the limit of positive weakly convergence sequence nonnegative?

Question

Let $\{u_n\}$ be a sequence of functions in $W^{1,p}(\mathbb{R}^N), p>1$, $u_n>0$ for all $n$, such that $u_n\rightharpoonup u$. Then $u\ge 0$. Is that true? If not, any counterexample? Thank you!

Answer 1

The set $$\{u \in W^{1,p} \;|\; u \ge 0 \; \text{a.e.}\}$$ is closed and convex, thus weakly closed.

Answer 2

Yes, this is true. Let $\phi\in C_c^\infty$ with $\phi\ge0$. Then $$ 0\le \int \phi u_m \to \int \phi u. $$ So, $ \int \phi u\ge0$ for all smooth and non-negative $\phi$. Hence $u\ge0$ a.e.