Infinitely many solutions for linearly coupled Schrödinger systems in ℝ3

• Published in: Mathematical methods in the applied sciences 2024-12, Vol.47 (18), p.13791-13812
• Main Authors: Wang, Lushun, Zeng, Dehua
• Format: Article
• Language: English
• Subjects: Arrays; linearly coupled Schrödinger system; Lyapunov–Schmidt reduction method; nonradial solutions; slow decaying rate; synchronized solutions
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.10239

In this paper, we consider the following linearly coupled nonlinear Schrödinger system: A −Δu+P(|y|)u=u3+λ(|y|)vinℝ3,−Δv+Q(|y|)v=v3+λ(|y|)uinℝ3,u,v∈H1(ℝ3),$$ \left\{\begin{array}{ll}-\Delta u+P\left(|y|\right)u={u}^3+\lambda \left(|y|\right)v& \kern0.4em \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}^3,\\ {}-\Delta v+Q\left(|y|\right)v={v}^3+\lambda \left(|y|\right)u& \kern0.4em \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}^3,\\ {}u,v\in {H}^1\left({\mathrm{\mathbb{R}}}^3\right),& \end{array}\right. $$ where P(|y|),Q(|y|)$$ P\left(|y|\right),Q\left(|y|\right) $$, and λ(|y|)$$ \lambda \left(|y|\right) $$ are radial, positive, and continuous and satisfying that lim|y|→∞P(|y|)=lim|y|→∞Q(|y|)=1,lim|y|→∞λ(|y|)=λ∈(0,1).$$ \underset{\mid y\mid \to \infty }{\lim }P\left(|y|\right)=\underset{\mid y\mid \to \infty }{\lim }Q\left(|y|\right)=1,\kern0.30em \underset{\mid y\mid \to \infty }{\lim}\lambda \left(|y|\right)=\lambda \in \left(0,1\right). $$ When λ(|y|)