Abstract
Our aim is to verify that the functional in the virial identity classifies the dynamics for nonlinear Schrödinger equations of local interactions. In particular, we give a condition under that there exist stable ground states. Our proof of this stability result is based on the ideas in Colin (Ann Inst H Pincaré 23:753–764, 2006) and Shatah (Math Phys 91:313–327, 1983). However, we emphasize that our argument does not use the strict convexity of the \(\dot{H}^{1}\)-norm of ground state with respect to \(\omega \): a key lemma is Lemma 4.8 below. Furthermore, we discuss the limiting profile of ground states (see Theorem 4.4).
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Akahori, T., Kikuchi, H. & Yamada, T. Virial functional and dynamics for nonlinear Schrödinger equations of local interactions. Nonlinear Differ. Equ. Appl. 25, 5 (2018). https://doi.org/10.1007/s00030-018-0497-7
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DOI: https://doi.org/10.1007/s00030-018-0497-7