Two-step minimization approach to an L^∞-constrained variational problem with a generalized potential

We study a variational problem on H¹( R) under an L^∞-constraint related to Sobolev-type inequalities for a class of generalized potentials, including Lᵖ-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, those 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.