Local and nonlocal problems regarding the Q-curvature and the Adams-Moser-Trudinger inequalities

We study the existence and classification of solutions to a Q-curvature problem in R^n with finite volume. Inspired by the previous works of Lin and Martinazzi in even dimension and Jin-Maalaoui-Martinazzi-Xiong in dimension three we classify all solutions in terms of their behavior at infinity. Extending the work of Wei-Ye we proved the existence of solution with prescribed volume and asymptotic behavior, under certain restrictions. In the case when the dimension n is bigger than four, we show that the volume of the conformal metric can be prescribed arbitrarily.

We also study a sharp Adams-Moser-Trudinger type inequality in a fractional settings. As an application, improving upon works of Adimurthi and Lakkis, we prove existence of solutions to a Moser-Trudinger equation.