On elliptic equations and systems with critical growth in dimension two

In this paper we consider nonlinear elliptic equations in bounded domains in R^2, with superlinear nonlinearities. In two dimensions the critical growth (i.e. maximal growth such that the problem can be treated variationally) is of exponential type, given by Pohozaev-Trudinger type inequalities. We discuss existence and nonexistence results related to critical growth for the equation and the system. The natural framework for such equations and systems are Sobolev spaces, which give in most cases an adequate answer concerning the maximal growth involved. However, we will see that for the system in dimension 2, the Sobolev embeddings are not sufficiently fine to capture the true maximal growths. We will show that working in Lorentz spaces gives better results.