 Constrained ground-state minimizer on a bounded domain

The infimum of $E(u)$ over $\Vert u{\Vert }_2=1$ is attained. A minimizing sequence is bounded in $H_0^1$ because $V$ is bounded below; weak compactness, Rellich strong $L^2$ convergence, and weak lower semicontinuity complete the direct-method argument.