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

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