Journal of the
Korean Mathematical Society JKMS

In this paper, we consider the problem of existence of conformal metrics with prescribed mean curvature on the unit ball of $\R^n$, $n\geq 3$. Under the assumption that the order of flatness at critical points of prescribed mean curvature function $H(x)$ is $\beta\in ]1,n-2]$, we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula.

Keywords: boundary mean curvature, variational method, loss of compactness, $\beta$-flatness condition, critical point at infinity

MSC numbers: 35B40, 53C21, 35J65