Schrödinger systems with critical coupling in $$ \mathbb {R}^N$$

Abstract This paper is concerned with the existence and multiplicity of positive solutions for nonlinear Schrödinger systems in $$ \mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> with a critical weak coupling. The existence of a positive least energy solution is also studied. The proofs of the main results rely on local and global minimization arguments and minimax methods applied to an associated functional restricted to the Nehari manifold. Appropriated estimates for the critical levels, based on the argument of Brezis and Nirenberg for the scalar problem, are a key ingredient to deal with the lack of compactness of the functional.