My question is how to obtain (strong) convergence in Frechet spaces $H^m_{loc}(\Omega)$, $m\ge1$ using the following result found in Lions (Perturbations Singulières dans les Problèmes aux Limites et Contrôle Optimal, p. 121), take m=1 for simplicity:
If
1) $u_\varepsilon\to u$ in $L^2_{loc}(\Omega)$,
2) $\varepsilon^{1/2}u_\varepsilon$ is bounded in $H^1_{loc}(\Omega)$,
3) $-\varepsilon \Delta u_\varepsilon+u_\varepsilon$ is bounded in $H^1_{loc}(\Omega)$.
Then,
(a) $\varepsilon^{1/2}u_\varepsilon$ is bounded in $H^2_{loc}(\Omega)$,
(b) $u_\varepsilon$ is bounded in $H^1_{loc}(\Omega)$.
Moreover, this implies that $u_\varepsilon\to u$ in $H^1_{loc}(\Omega)$.
My problem is with the last assertion, I do see how to get the convergence in $H^1_{loc}(\Omega)$
It is false. If you take $v_n=\frac1{n^{1/2}}\chi_{[0,1/n]}$ and $u_n(x)=\int_0^x v_n(s)\,dx$, you have that $v_n$ is bounded in $L^2(\mathbb{R})$ with norm $1$ and so $u_n$ is bounded in $H^1_{loc}(\mathbb{R})$ but $u'_n=v_n$ does not converge in the compact set $[0,1]$ since it converges to zero but its $L^2$ norm in $[0,1]$ is $1$.
The sequence in Lions book has several additional properties.