“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 $\mathrm {max}{p\in M}f\_0(p)=0$, 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 $u^{\lambda}\neq u\_{\lambda}$ 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 $\lambda\downarrow 0$.