Blow-up analysis for solutions to Neumann boundary value problem
The authors study the Liouville equation \[ \begin{cases} -\Delta_g u = 2 e^{2u} - K_g & \text{ in } M^o,\\ \frac{\partial u}{\partial n} = c e^{u} - h_g & \text{ on } \partial M, \end{cases} \] where \((M,g)\) is a compact two-dimensional surface with smooth boundary \(\partial M\), \(K_g\) is the Gaussian curvature of \(M\), \(h_g\) is the geodesic curvature of \(\partial M\), \(\Delta_g\) is the Laplace-Beltrami operator with respect to \(g\), and \(c\) is a given constant. The authors prove that the blow-up set of solution sequences \((u_n)_{n \in \mathbb{N}}\) is finite and determine the possible blow-up values and the blow-up profile. Important ingredients of the proofs are adaptations of the methods from [\textit{H. Brezis} and \textit{F. Merle}, Commun. Partial Differ. Equations 16, No. 8--9, 1223--1253 (1991; Zbl 0746.35006); \textit{D. Bartolucci} et al., Commun. Partial Differ. Equations 29, No. 7--8, 1241--1265 (2004; Zbl 1062.35146)].
scientific article; zbMATH DE number 6391294