The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications
Lewin, Mathieu; Rota Nodari, Simona (2020), The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications, Calculus of Variations and Partial Differential Equations, 59, 197, p. 1-53. 10.1007/s00526-020-01863-w
TypeArticle accepté pour publication ou publié
Journal nameCalculus of Variations and Partial Differential Equations
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Rota Nodari, Simona
Institut de Mathématiques de Bourgogne [Dijon] [IMB]
Abstract (EN)In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form Δu+g(u)=0. Our result applies in particular to the double power non-linearity where g(u)=uq−up−μu for p>q>1 and μ>0, which we discuss with more details. In this case, the non-degeneracy of the unique solution uμ allows us to derive its behavior in the two limits μ→0 and μ→μ∗ where μ∗ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the L2 mass of uμ in terms of μ, which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of uμ.
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