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Existence of the ground state for the NLS with potential on graphs. (arXiv:1707.07326v2 [math-ph] UPDATED)
来源于:arXiv
We review and extend several recent results on the existence of the ground
state for the nonlinear Schr\"odinger (NLS) equation on a metric graph. By
ground state we mean a minimizer of the NLS energy functional constrained to
the manifold of fixed $L^2$-norm. In the energy functional we allow for the
presence of a potential term, of delta-interactions in the vertices of the
graph, and of a power-type focusing nonlinear term. We discuss both subcritical
and critical nonlinearity. Under general assumptions on the graph and the
potential, we prove that a ground state exists for sufficiently small mass,
whenever the constrained infimum of the quadratic part of the energy functional
is strictly negative. 查看全文>>