Lorentz-Sobolev spaces and systems of Schroedinger equations in R^N

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 Schroedinger 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 Pohozaev-Trudinger-Moser inequality and still preserving a variational structure.