Least energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}^n}\). (English) Zbl 1147.35098
The paper is concerned with the following system in \(\mathbb{R}^n\), \(n\leq 3\):
\[
\Delta u_1 - u_1+\mu_1 u^3_1+\beta u_1 u^2_2=0, \quad \Delta u_2 -\lambda u_2+\mu_2 u^3_2+\beta u^2_1 u_2=0, \qquad u_1, u_2\in H^1(\mathbb{R}^n).
\]
\(\lambda\geq1\), \(\mu_1, \mu_2>0\) and \(\beta\in\mathbb{R}\) are parameters. One is interested in solutions \(u=(u_1, u_2)\) with both \(u_1\neq 0\) and \(u_2\neq 0\). The problem is of variational nature, solutions can be obtained as critical points of the associated energy functional \(E:H=H^1(\mathbb{R}^n)\times H^1(\mathbb{R}^n)\to\mathbb{R}\). A natural constraint is the set
\[ \mathcal{N}=\{u\in H: u_1\neq 0,\;u_2\neq 0,\;E'(u)[u_1,0]=0=E'(u)[0,u_2]\}. \]
The main results of the paper yield parameter ranges for \(\lambda,\beta,\mu_1, \mu_2\) such that \(E\) achieves its infimum on \(\mathcal{N}\) or on the set of radial functions in \(\mathcal{N}\), and parameter ranges where the infimum is not achieved. This improves earlier results of T.-C. Lin and J. Wei [Commun. Math. Phys. 255, No. 3, 629–653 (2005; Zbl 1119.35087)].
\(\lambda\geq1\), \(\mu_1, \mu_2>0\) and \(\beta\in\mathbb{R}\) are parameters. One is interested in solutions \(u=(u_1, u_2)\) with both \(u_1\neq 0\) and \(u_2\neq 0\). The problem is of variational nature, solutions can be obtained as critical points of the associated energy functional \(E:H=H^1(\mathbb{R}^n)\times H^1(\mathbb{R}^n)\to\mathbb{R}\). A natural constraint is the set
\[ \mathcal{N}=\{u\in H: u_1\neq 0,\;u_2\neq 0,\;E'(u)[u_1,0]=0=E'(u)[0,u_2]\}. \]
The main results of the paper yield parameter ranges for \(\lambda,\beta,\mu_1, \mu_2\) such that \(E\) achieves its infimum on \(\mathcal{N}\) or on the set of radial functions in \(\mathcal{N}\), and parameter ranges where the infimum is not achieved. This improves earlier results of T.-C. Lin and J. Wei [Commun. Math. Phys. 255, No. 3, 629–653 (2005; Zbl 1119.35087)].
Reviewer: Thomas J. Bartsch (Gießen)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q51 | Soliton equations |
| 35J50 | Variational methods for elliptic systems |
Citations:
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