A natural constraint approach to normalized solutions of nonlinear Schr\"odinger equations and systems

By Thomas Bartsch, Nicola Soave. 2016

317 citations recorded by the source corpus.

We prove existence of normalized solutions to \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 & \text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v & \text{in $\mathbb{R}^3$}\int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2 \end{cases} \] for any $\mu_1,\mu_2,a_1,a_2>0$ and $\beta<0$ prescribed. The approach is based upon the introduction of a natural constraint associated to the problem. Our method can be adapted to the scalar NLS equation with normalization constraint, and leads to alternative and simplified proofs to some results already available in the literature.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e70e57367b43f9e61df8e6d78863b2c10eb31039

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Citations in the corpus

Multiple positive normalized solutions for nonlinear Schrödinger systems (2017, background): “Let us also mention the recent work [6] where, under the condition that β < 0, a positive solution is obtained under quite general assumptions, see also [7] for a multiplicity result.”
Normalized solutions to mass supercritical Schrödinger equations with negative potential (2021, background): “After [22], only recently more general autonomous equations and systems have been considered, also refining and developing this initial strategy, see [4, 6, 8, 5, 17, 9, 21, 34, 35] and references therein.”
Normalized solutions of $L^2$-supercritical NLS equations on compact metric graphs (2022, methodology): “We point out that the techniques based on scalings, usually employed in the Euclidean setting (see [21] or [7, 20]), do not work, since G is not scale invariant.”
Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case (2019, unknown): “[7, 10, 33, 46].”
Normalized ground states for the NLS equation with combined nonlinearities (2018, unknown): “Among the other contributions, we refer in particular to [8, 11, 31], which strongly inspired many techniques used here.”
Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems (2022, methodology): “We refer the reader to [7, 26, 27, 28, 29, 30, 39, 40, 43] for scalar equations, [8, 9, 10, 11, 14, 21, 22, 31] for systems of two equations, [35] for systems of k equations.”
Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities (2021, background): “The new constraint, which is introduced by Bartsch and Soave in [9] for the general problem (1.”