If $f_j \rightharpoonup f$ weakly in $W^{1,p}$ then $f_j \to f$ strongly in $L^p$?

Question

Suppose $1<p<\infty$ and $\Omega$ is an open bounded set in $\mathbb R^n$ with nice boundary (say Lipschitz or even better). Let $(f_j)_j \subset W^{1,p}(\Omega)$ s.t. $f_j \rightharpoonup f$ weakly in $W^{1,p}(\Omega)$.

Is it true that $f_j \to f$ strongly in $L^p(\Omega)$?

For sure it is true that $f_j \rightharpoonup f$ and $\nabla f_j \rightharpoonup\nabla f$. Moreover, we should have the strong convergence of a subsequence thanks to reflexivity: $(f_j)_j$ is bounded hence is has a strong convergent subsequence in $L^p(\Omega)$ because the embedding $W^{1,p} \to L^p$ is (always) compact.

Thanks.

Answer 1

Hint: prove the following topological result

Assume that $\Omega$ is a metric space and $x_n\in\Omega$ is a sequence. Suppose that every subsequence of $x_n$ has a further subsequence, which converges to some fixed limit $x\in \Omega$. Then, $$x_n\to x$$