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July/August 2022 Normalized ground states to a cooperative system of Schrödinger equations with generic $L^2$-subcritical or $L^2$-critical nonlinearity
Jacopo Schino
Adv. Differential Equations 27(7/8): 467-496 (July/August 2022).

Abstract

We look for ground state solutions to the Schrödinger-type system\[\begin{cases} -\Delta u_j+\lambda_ju_j=\partial_jF(u)\\ \displaystyle \int_{\mathbb R^N}u_j^2\,dx=a_j^2\\ (\lambda_j,u_j)\in\mathbb R\times H^1 (\mathbb R^N)\end{cases}j\in\{1,\dots,M\}\]with $N,M \ge 1$, where $a=(a_1,\dots,a_M)\in ( 0,\infty ) ^M$ is prescribed and $(\lambda,u)=(\lambda_1,\dots,\lambda_M,u_1,\dots u_M)$ is the unknown. We provide generic assumptions about the nonlinearity $F$ which correspond to the $L^2$-subcritical and $L^2$-critical cases, i.e., when the energy is bounded from below for all or some values of $a$. Making use of a recent idea, we minimize the energy over the constraint $\{\left|u_j\right|_{L^2} \le a_j \text{ for all } j \}$ and then provide further assumptions that ensure $|u_j|_{L^2}=a_j$.

Citation

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Jacopo Schino. "Normalized ground states to a cooperative system of Schrödinger equations with generic $L^2$-subcritical or $L^2$-critical nonlinearity." Adv. Differential Equations 27 (7/8) 467 - 496, July/August 2022.

Information

Published: July/August 2022
First available in Project Euclid: 26 April 2022

Subjects:
Primary: 35J20 , 35Q40 , 35Q60 , 78A25

Rights: Copyright © 2022 Khayyam Publishing, Inc.

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Vol.27 • No. 7/8 • July/August 2022
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