Let $u_n $ be a sequence in $H^1 (\Omega,\mathbb{C})$ where $\Omega \subset \mathbb{R} ^N$ is bounded. Assume that we know that all the functions $u_n$ are smooth and $\Vert u_n \Vert _{C^m} \leq K(m,N)$ and moreover $\vert u_n \vert \leq K_1$ and $\vert \nabla u_n \vert\leq K_2$ where $K_i>0$ is a constant (not depending on $n$). I wonder if $\lbrace u_n\rbrace$ is strongly convergent in $H^1 (\Omega)$.
Since I have the uniform bounds $\vert u_n \vert \leq K_1$ and $\vert \nabla u_n \vert\leq K_2$, a good attempt is to use Dominated Convergence Theorem. But then I also need convergence pointwise of $u_n$ to some $u\in H^1(\Omega,\mathbb{C})$. If I show that $u_n$ converges pointwise to $u$ then, by DCT in $L^p$ spaces, $u_n \rightarrow u$ in $L^2$. Similarly, $\nabla u_n \rightarrow \nabla u$ in $L^2$. So, therefore $u_n \rightarrow u$ in $H^1$. I think that this reasoning is right... so I only have to prove that $u_n$ and $\nabla u_n$ converge pointwise. Any help to show that?
Thank you in advance.
PS: I know that bounded sequences $u_n$ in $H^1$ admits a weak limit $u$, and so $u_n$ converges pointwise almost everywhere to $u$. But I don't know if my sequence is bounded in $H^1$, do I?