Positive solutions of the Gross-Pitaevskii equation for energy critical and supercritical nonlinearities

Autor:	Pelinovsky, D. E., Wei, J., Wu, Y.
Rok vydání:	2022
Předmět:	Mathematics - Analysis of PDEs Mathematical Physics Mathematics - Spectral Theory Nonlinear Sciences - Pattern Formation and Solitons
Druh dokumentu:	Working Paper
DOI:	10.1088/1361-6544/acd90a
Popis:	We consider positive and spatially decaying solutions to the Gross-Pitaevskii equation with a harmonic potential. For the energy-critical case, there exists a ground state if and only if the frequency belongs to (1,3) in three dimensions and in (0,d) in d dimensions. We give a precise description on asymptotic behaviors of the ground state up to the leading order term for different values of d. For the energy-supercritical case, there exists a singular solution for some frequency in (0,d). We compute the Morse index of the singular solution in the class of radial functions and show that the Morse index is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for nonlinearity powers not large enough and is equal to 1 in the monotone case for nonlinearity power sufficiently large. Comment: 28 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2207.10145 Zobrazit plný text záznamu View this record from Arxiv