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.
Non-uniqueness of normalized NLS ground states on polygons with homogeneous Neumann boundary conditions
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We provide a non-uniqueness result for normalized ground states of nonlinear Schr\"odinger equations with pure power nonlinearity on polygons with homogeneous Neumann boundary conditions, defined as global minimizers of the associated energy functional among functions with prescribed mass. Precisely, for nonlinearity powers slightly smaller than the $L^2$-critical exponent, we prove that there always exists at least one value of the mass for which normalized ground states are not unique.
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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.