Ground-state eigenfunction of a confining Schrödinger operator

The minimum ofsubject to $\Vert u{\Vert }_2=1$ is attained because ${\Sigma }_V$ embeds compactly into $L^2$. Replacing a minimizer by its absolute value does not increase its energy. The Euler-Lagrange equation therefore produces a nonnegative eigenfunction of $-\Delta +V$ for its lowest eigenvalue.