= Solution
Weak convergence $u_n\rightharpoonup u$ in $H_0^1(\Omega)$ implies
$$
Du_n\rightharpoonup Du
$$
in $L^2$. The $L^2$ norm is weakly lower semicontinuous, so
$$
\int_\Omega|Du|^2
\leq\liminf_n\int_\Omega|Du_n|^2.
$$
The <Rellich-Kondrashov compactness theorem for H01> upgrades the weak convergence to
$$
u_n\longrightarrow u
$$
strongly in $L^2(\Omega)$. Indeed, compactness gives this along every subsequence after passage to a further subsequence, and the weak limit uniquely identifies every such strong limit as $u$. Consequently,
$$
\begin{aligned}
\left|\int_\Omega V(u_n^2-u^2)\right|
&\leq\lVert V\rVert_\infty
\lVert u_n-u\rVert_2
\bigl(\lVert u_n\rVert_2+\lVert u\rVert_2\bigr)\\
&\longrightarrow0.
\end{aligned}
$$
Combining the two terms proves the <weak lower semicontinuity of a bounded-domain Schrodinger energy>:
$$
\boxed{E(u)\leq\liminf_nE(u_n)}.
$$
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