A construction of peak solutions by a local mountain pass approach for a nonlinear Schrödinger system with three wave interaction

Yuki Osada () and Yohei Sato ()
Additional contact information
Yuki Osada: Saitama University
Yohei Sato: Saitama University

Partial Differential Equations and Applications, 2025, vol. 6, issue 1, 1-26

Abstract: Abstract In this paper, we consider the following nonlinear Schrödinger system with three wave interaction: $$\begin{aligned} {\left\{ \begin{array}{ll} - \varepsilon ^2 \Delta u_1 + V_1(x) u_1 = \mu _1 |u_1|^{p-1} u_1 + \alpha u_2 u_3,\quad \text {in}\ \mathbb {R}^N,\\ - \varepsilon ^2 \Delta u_2 + V_2(x) u_2 = \mu _2 |u_2|^{p-1} u_2 + \alpha u_1 u_3,\quad \text {in}\ \mathbb {R}^N,\\ - \varepsilon ^2 \Delta u_3 + V_3(x) u_3 = \mu _3 |u_3|^{p-1} u_3 + \alpha u_1 u_2,\quad \text {in}\ \mathbb {R}^N, \end{array}\right. } \end{aligned}$$ - ε 2 Δ u 1 + V 1 ( x ) u 1 = μ 1 | u 1 | p - 1 u 1 + α u 2 u 3 , in R N , - ε 2 Δ u 2 + V 2 ( x ) u 2 = μ 2 | u 2 | p - 1 u 2 + α u 1 u 3 , in R N , - ε 2 Δ u 3 + V 3 ( x ) u 3 = μ 3 | u 3 | p - 1 u 3 + α u 1 u 2 , in R N , where $$N \le 5$$ N ≤ 5 , $$1 0$$ ε > 0 , $$V_j(x)>0$$ V j ( x ) > 0 , $$\mu _j > 0\ (j=1,2,3)$$ μ j > 0 ( j = 1 , 2 , 3 ) and $$\alpha > 0$$ α > 0 . We construct a peak solution that is concentrating at a local minimum point of a function $$c(V_1(x),V_2(x),V_3(x))$$ c ( V 1 ( x ) , V 2 ( x ) , V 3 ( x ) ) . Here $$c(\lambda _1,\lambda _2,\lambda _3)$$ c ( λ 1 , λ 2 , λ 3 ) is a mountain pass value of the following limit system $$\begin{aligned} {\left\{ \begin{array}{ll} - \Delta v_1 + \lambda _1 v_1 = \mu _1 |v_1|^{p-1} v_1 + \alpha v_2 v_3\quad \text {in}\ \mathbb {R}^N,\\ - \Delta v_2 + \lambda _2 v_2 = \mu _2 |v_2|^{p-1} v_2 + \alpha v_1 v_3\quad \text {in}\ \mathbb {R}^N,\\ - \Delta v_3 + \lambda _3 v_3 = \mu _3 |v_3|^{p-1} v_3 + \alpha v_1 v_2\quad \text {in}\ \mathbb {R}^N. \end{array}\right. } \end{aligned}$$ - Δ v 1 + λ 1 v 1 = μ 1 | v 1 | p - 1 v 1 + α v 2 v 3 in R N , - Δ v 2 + λ 2 v 2 = μ 2 | v 2 | p - 1 v 2 + α v 1 v 3 in R N , - Δ v 3 + λ 3 v 3 = μ 3 | v 3 | p - 1 v 3 + α v 1 v 2 in R N . When $$p\in (1,2)$$ p ∈ ( 1 , 2 ) , this limit system does not necessarily have a ground state. Hence a key of the construction is to use a local mountain pass approach.

Keywords: Coupled nonlinear Schrödinger equations; Schrödinger systems; Three wave interaction; Peak solution; Local mountain pass; 35J57; 35J50; 35J91 (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s42985-025-00312-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:6:y:2025:i:1:d:10.1007_s42985-025-00312-7

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-025-00312-7

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().