The authors investigate the symmetry properties of extremal functions for certain maximization problems related to Moser-Trudinger inequalities. The symmetry of maximizers of a Hénon type functional in dimension two is considered, and the maximization problem
\[ S_{\alpha , \gamma}=\sup_{\substack{ u\in H^1(\Omega)\\ \|u\|\leq1 }} \int_\Omega (e^{\gamma u^2}-1)|x|^{\alpha}\,dx, \]
is studied, where \(\Omega\) is the unit ball of \(\mathbb R^2\), \(\alpha\) and \(\gamma\) are positive real numbers and \(\|u\|\) denotes the usual \(H^1(\Omega)\) norm. Moreover, the limit problem
\[ S_{\gamma}= \sup_{\substack{ u\in H^1(\Omega)\\ \|u\|\leq1 }} \int_{\partial\Omega} (e^{\gamma u^2}-1) \,d{\sigma}, \]
is investigated, which is the main ingredient to describe the behavior of maximizers as \(\alpha\to\infty\). The limit problem as \(\alpha\to 0\) and the properties of its maximizers are also discussed.
The main results in this paper are the Moser-Trudinger trace inequality and the picture of the symmetry properties of solutions for different values of \(\gamma\). This interesting survey also incudes some existence and unique results, the asymptotic analysis for the maximization problem, a first symmetry breaking result, Moser-Trudinger type inequalities with weight for the limit problem, a conjecture to deserve further studies. These are the extension to dimension two of the corresponding results in dimension \(n\geq 3\).