Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures

Abstract

Consider a compact Riemannian surface $(M,g)$ with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions $f$ in $M$ and $h$ in $\partial M$ with $\max f= \max h= 0$, under a suitable condition on the maximum points of $f$ and $h$, we prove that for sufficiently small positive constants $λ$ and $μ$, there exist at least two distinct conformal metrics $g_{λ,μ}=e^{2u_{μ,λ}}g$ and $g^{λ,μ}=e^{2u^{μ,λ}}g$ with prescribed sign-changing Gaussian and geodesic curvature equal to $f + μ$ and $h + λ,$ respectively. Additionally, we employ the method used in Borer et al. (2015) to study the blowing up behavior of the large solution $u^{μ,λ}$ when $μ\downarrow 0$ and $λ\downarrow 0$. Finally, we derive a new Liouville-type result for the half-space, eliminating one of the potential blow-up profiles.