Normalized solutions for nonlinear Schrödinger systems

We consider the existence of normalized solutions in H 1 (ℝ N ) × H 1 (ℝ N ) for systems of nonlinear Schr¨odinger equations, which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz, one is led to coupled systems of elliptic equations of the form and we are looking for solutions satisfying where a 1 > 0 and a 2 > 0 are prescribed. In the system, λ 1 and λ 2 are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e. , with positive constants β , μ i , p i , r i . The exponents are Sobolev subcritical but may be L 2 -supercritical. Our main result deals with the case in which in dimensions 2 ≤ N ≤ 4. We also consider the cases in which all of these numbers are less than 2 + 4 /N or all are bigger than 2 + 4 /N .