Energy local minimizers with prescribed L2 norm exist in the mass-supercritical NLS on R^N x M^k; for small mass they are the Euclidean ground states, and in certain cases they become nontrivial along M^k.
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Energy local minimizers for the nonlinear Schr\"{o}dinger equation on product spaces
Energy local minimizers with prescribed L2 norm exist in the mass-supercritical NLS on R^N x M^k; for small mass they are the Euclidean ground states, and in certain cases they become nontrivial along M^k.