Soluzioni di equazioni di Liouville a profilo non banale (Solutions of Liouville equations with non-trivial profile)

Liouville equations have been widely studied for more than a century. In particular, the interest in this class of PDEs renewed during the last three decades, after the introduction of the so-called Q-curvature and the discovery that they are intimately related to several fundamental concepts both in Analysis and in Geometry. In this work, we will show the existence of a class of non-trivial solutions of the 2D Liouville equation with infinite volume, employing basic tools of bifurcation theory. Using some more advanced techniques of bifurcation theory and Morse theory, we will also lay the groundwork for the study of the same problem in dimension 4.