|
Session 38. Variational Methods in Nonlinear Analysis
|
Minimal energy solutions for repulsive nonlinear Schrödinger systems |
Rainer Mandel, Karlsruhe Institute of Technology, Germany
|
|
|
We establish existence and nonexistence results concerning fully nontrivial minimal energy
solutions of the nonlinear Schrödinger system
\begin{align*}
\begin{aligned}
-\Delta u + \; u &= |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad\text{in }\mathbb{R}^n, \\
-\Delta v + \omega^2 v &= |v|^{2q-2}v + b|u|^q|v|^{q-2}v\quad $\,$ \text{in }\mathbb{R}^n.
\end{aligned}
\end{align*}
We consider the repulsive case \(b<0\) and assume that the exponent \(q\) satisfies \(1<q<\frac{n}{n-2}\) in case
\(n\geq 3\) and \(1<q<\infty\) in case \(n=1\) or \(n=2\). For space dimensions \(n\geq 2\) and arbitrary \(b<0\) we prove the
existence of fully nontrivial nonnegative solutions which converge to a solution of some optimal partition
problem as \(b\to -\infty\). In case \(n=1\) we prove that minimal energy solutions exist provided the coupling
parameter \(b\) has small absolute value whereas fully nontrivial solutions do not exist if \(1<q\leq 2\) and
\(b\) has large absolute value. This generalizes the existence results found in [1].
|
|
|
References-
B. Sirakov: Least energy solitary waves for a system of nonlinear Schrödinger
equations in \(R^n\), Comm. Math. Phys. 271 (2007), 199-221.
- R. Mandel: Minimal energy solutions for repulsive nonlinear Schrödinger systems, http://arxiv.org/abs/1303.4521
|
|
| Print version |
|
