Multi-peak solutions for coupled nonlinear Schrödinger systems in low dimensions. (English) Zbl 1512.35035
This paper investigates the solutions to the following nonlinear Schrödinger system
\begin{align*}
-\varepsilon^2 \Delta u+P(x)u&=\mu_1u^p+\beta u^{\frac{p-1}2}v^{\frac{p+1}2},\\
-\varepsilon^2 \Delta v+Q(x)v&=\mu_2v^p+\beta u^{\frac{p+1}2}v^{\frac{p-1}2},
\end{align*}
in \(\mathbb{R}^N\), where \(3<p<+\infty\), \(N=1,2\), \(\varepsilon>0\) is a small parameter, the potentials \(P,\,Q\) satisfy \(0 < P_0 \le P(x) \le P_1\) and \(Q(x)\) satisfies \(0 < Q_0 \le Q(x) \le Q_1\). By using the Lyapunov-Schmidt reduction argument, this paper constructs the solutions with \(k\) peaks for attractive and repulsive cases. This type of results extend the main results established by S. Peng and Z.-q. Wang [Arch. Ration. Mech. Anal. 208, No. 1, 305–339 (2013; Zbl 1260.35211)] and H. Pi and S. Peng [Discrete Contin. Dyn. Syst. 36, No. 4, 2205–2227 (2016; Zbl 1327.35038)], where the the case \(N=3, p=3\) was considered.
Reviewer: Shuangjie Peng (Wuhan)
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35J47 | Second-order elliptic systems |
| 35J61 | Semilinear elliptic equations |
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