“Large” conformal metrics of prescribed Gauss curvature on surfaces of higher genus

Abstract

Let $(M,g_0)$ be a closed Riemann surface $(M,g_0)$ of genus $\gamma (M)>1$ and let $f_0$ be a smooth, non-constant function with ${\max }_{p\in M}{f}_0(p)=0$, all of whose maximum points are non-degenerate. As shown in [12] for sufficiently small $\lambda >0$ there exist at least two distinct conformal metrics $g_{\lambda }=e^{2u_{\lambda }}{g}_0$, $g^{\lambda }=e^{2u^{\lambda }}{g}_0$ of Gauss curvature $K_{g_{\lambda }}=K_{g^{\lambda }}=f_0+\lambda$, where $u_{\lambda }$ is a relative minimizer of the associated variational integral and where $u^{\lambda }\ne u_{\lambda }$ is a further critical point not of minimum type. Here, by means of a more refined mountain-pass technique we obtain additional estimates for the "large" solutions $u^{\lambda }$ that allow to characterize their "bubbling behavior" as $\lambda \downarrow 0$.