A natural constraint approach to normalized solutions of nonlinear   Schr\"odinger equations and systems
  We prove existence of normalized solutions to \[ \begin{cases} -\Delta u - \lambda_1 u = \mu_1 u^3+ \beta u v^2 & \text{in $\mathbb{R}^3$} -\Delta v- \lambda_2 v = \mu_2 v^3 +\beta u^2 v & \text{in $\mathbb{R}^3$}\int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2 \end{cases} \] for any $\mu_1,\mu_2,a_1,a_2>0$ and $\beta<0$ prescribed. The approach is based upon the introduction of a natural constraint associated to the problem.   Our method can be adapted to the scalar NLS equation with normalization constraint, and leads to alternative and simplified proofs to some results already available in the literature. 
Thomas Bartsch and Nicola Soave
math.AP