Orbital stability of ground states for a Sobolev critical Schrödinger equation
By Louis Jeanjean, Jacek Jendrej, Thanh Trung Le, Nicola Visciglia. Journal des Mathématiques Pures et Appliquées, 2020
146 citations recorded by the source corpus.
We study the existence of ground state standing waves, of prescribed mass, for the nonlinear Schrodinger equation with mixed power nonlinearities \begin{equation*} i \partial_t v + \Delta v + \mu v |v|^{q-2} + v |v|^{2^* - 2} = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R}^N, \end{equation*} where $N \geq 3$, $v: \mathbb{R} \times \mathbb{R}^N \to \mathbb{C}$, $\mu > 0$, $2 < q < 2 + 4/N $ and $2^* = 2N/(N-2)$ is the critical Sobolev exponent. We show that all ground states correspond to local minima of the associated Energy functional. Next, despite the fact that the nonlinearity is Sobolev critical, we show that the set of ground states is orbitally stable. Our results settle a question raised by N. Soave [35].
Subfields: Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2873b7ae08ff52bcabda51b306f4eb1f53825208
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- Normalized solutions to a class of Kirchhoff equations with Sobolev critical exponent (2021, background): “5) are summarized in the following table: N μ p and q classifications of solutions references N ≥ 1 μ > 0 2 < q < p ≤ 2 + 4 N a global minimizer [7, 27] N ≥ 1 μ < 0 2 < q ≤ 2 + 4 N < p < 2∗ a Mountain Pass solution [27] N ≥ 1 μ > 0 2 < q < 2 + 4 N < p < 2∗ a local minimizer; a Mountain Pass solution”