$(u_n)$ is bounded in $H^1(\mathbb{R}^N)$, some results about the convergence.

Question

If $(u_n)$ is bounded in the Hilbert space $H^1(\mathbb{R}^N)$, we have that, up to a subsequence, \begin{eqnarray} &&u_n \rightharpoonup u\ \mbox{ weakly in }H^1(\mathbb{R}^N),\\ &&u_n \rightarrow u \mbox{ in }L_{loc}^s(\mathbb{R}^N)\mbox{ for all } 2\leq s< 2^*,\\ &&u_n(x) \rightarrow u(x) \mbox{ for a.e. }x \in \mathbb{R}^N,\\ && \nabla u_n(x) \rightarrow \nabla u(x)\mbox{ for a.e. } x \in \mathbb{R}^N,\\ &&\nabla u_n \rightarrow \nabla u \mbox{ in } \bigg(L_{loc}^s(\mathbb{R}^N)\bigg)^N\mbox{ for all } 2\leq s<2^*.\\ \end{eqnarray}

Are these results above right?

Answer 1

1. True, by weak compactness
2. True, by 1) and Rellich-Kondrachov
3. True, by 2) and a standard $L^p$ space lemma
4. False: $\exp(inx)$ converge weakly to zero on $[0,1]$ but the derivatives do not converge anywhere a.e.
5. False: the gradients need not be locally in $L^s$ for $s>2$.