  We study a variational problem on $H^1({\mathbb R})$ under an
$L^\infty$-constraint related to Sobolev-type inequalities for a class of
generalized potentials, including $L^p$-potentials, non-positive potentials,
and signed Radon measures. We establish various essential tools for this
variational problem, including the decomposition principle, the comparison
principle, and the perturbation theorem, which are the basis of the two-step
minimization method. As for their applications, we present precise results for
minimizers of minimization problems, such as the study of potentials of Dirac's
delta measure type and the analysis of trapped modes in potential wells.
