Sharp Onofri trace inequality on the upper half space and quasi-linear Liouville equation with Neumann boundary

In this paper, we establish a sharp Onofri trace inequality on the upper half space $\overline{{\mathbb{R}}_{+}^n}(n\ge 2)$ by considering the limiting case of Sobolev trace inequality and classify its extremal functions on a suitable weighted Sobolev space. For this aim, by the Serrin-Zou type identity and the Pohozaev type identity, we show the classification of the solutions for a quasi-linear Liouville equation with Neumann boundary which is closely related to the Euler-Lagrange equation of the Onofri trace inequality and it has independent research value. The regularity and asymptotic estimates of solutions to the above equation are essential to discuss.