Lorentz-Sobolev spaces and systems of Schrödinger equations in RN

We study the connection between the improvement of limiting Sobolev's embeddings within the context of Lorentz spaces and the variational approach to systems of nonlinear Schrödinger equations. We show that Lorentz-Sobolev spaces appear as a natural function space domain for the related energy functional. Moreover, in this framework the nonlinearity may exhibit a supercritical growth with respect to the maximal growth prescribed by the Pohožaev-Trudinger-Moser inequality and still preserving a variational structure.