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Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2020

The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications

Résumé

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $\Delta u+g(u)=0$. Our result applies in particular to the double power non-linearity where $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu>0$, which we discuss with more details. In this case, the non-degeneracy of the unique solution $u_\mu$ allows us to derive its behavior in the two limits $\mu\to0$ and $\mu\to\mu_*$ where $\mu_*$ 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 $L^2$ mass of $u_\mu$ in terms of $\mu$, 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 $u_\mu$.

Dates et versions

hal-02792656 , version 1 (05-06-2020)

Identifiants

Citer

Mathieu Lewin, Simona Rota Nodari. The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications. Calculus of Variations and Partial Differential Equations, 2020, 59, pp.197. ⟨10.1007/s00526-020-01863-w⟩. ⟨hal-02792656⟩
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