Minimal energy solutions for cooperative nonlinear Schrödinger systems


DOI: 10.1007/s00030-014-0281-2

Cite this article as:
Mandel, R. Nonlinear Differ. Equ. Appl. (2015) 22: 239. doi:10.1007/s00030-014-0281-2


We prove sharp existence and nonexistence results for minimal energy solutions of the nonlinear Schrödinger system
$$\left.\begin{array}{ll}\quad -\Delta u + u = |u|^{2q-2}u + b|u|^{q-2}u|v|^q \quad {\rm in} \, \mathbb{R}^{n},\\ -\Delta v + \omega^2 v = |v|^{2q-2}v + b|u|^q|v|^{q-2}v \quad {\rm in} \, \mathbb{R}^{n}\end{array}\right.$$
in the cooperative and subcritical case \({b > 0, 1 < q < \frac{n}{(n-2)_+}}\) . The proofs are accomplished by minimizing the Euler functional of (1) over the two associated Nehari manifolds. In the special case \({1 < q < 2}\) we find that a positive solution of (1) with minimal energy among all nontrivial solutions exists if and only if b > 0.

Mathematics Subject Classification (2010)

Primary: 35J50 35J57 


Variational methods for elliptic systems Nonlinear Schrödinger systems 

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsKarlsruhe Institute of Technology (KIT)KarlsruheGermany