For supercritical power combinations, the NLS with a small potential on large bounded domains admits a negative-energy minimizer and a positive-energy mountain pass solution; in the Sobolev critical case, a ground state and a high-energy solution exist for small mass.
Agrawal, Nonlinear fiber optics, Springer (2000)
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Normalized solutions for NLS equations with potential on bounded domains: Ground states and multiplicity
For supercritical power combinations, the NLS with a small potential on large bounded domains admits a negative-energy minimizer and a positive-energy mountain pass solution; in the Sobolev critical case, a ground state and a high-energy solution exist for small mass.