If $f_k \rightarrow 0$ weakly, $f_k > 0$ a.e. then $f_k\rightarrow 0$ strongly?

Question

Let $\Omega$ be a bounded open, connected set in $\mathbb{R}^n$, $1<p<\infty$, and $\{f_k\}_{k=1}^\infty\subset L^p(\Omega)$ be a sequence of positive functions such that $f_k \to 0$ weakly in $L^p(\Omega)$. Does it imply that $f_k \to 0$ strongly in $L^p(\Omega)$?

Thanks in advance.

Answer 1

No. Take $\Omega = (0,1)$ and $f_k = k^{1/p} \, \chi_{(0,1/k)}$.