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Stationary solutions of semilinear Schr\"odinger equations with trapping potentials in supercritical dimensions

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arxiv 2210.14014 v1 pith:BWAN6OID submitted 2022-10-25 math-ph math.APmath.CAmath.MP

Stationary solutions of semilinear Schr\"odinger equations with trapping potentials in supercritical dimensions

classification math-ph math.APmath.CAmath.MP
keywords equationsschrdimensionsmethododingerpotentialssupercriticalapproach
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Nonlinear Schr\"odinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schr\"odinger-Newton and Gross-Pitaevskii equations with harmonic potentials.

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