On Elliptic Equations and Systems with Critical Growth in Dimension Two

We consider nonlinear elliptic equations of the form $-\Delta u=g(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, and Hamiltonian-type systems of the form $-\Delta u=g(v)$ in $\Omega$, $-\Delta v=f(u)$ in $\Omega$, $u=0$ and $v=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\mathbb R^2$ and $f,g\in C(\mathbb R)$ are superlinear nonlinearities. In two dimensions the maximal growth ($={}$critical growth) of $f$ and $g$ (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 the critical growth for the equation and the system. A natural framework for such equations and systems is given by Sobolev spaces, which provide 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.