The $$C^0$$-convergence at the Neumann boundary for Liouville equations

Abstract

In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li (Commun Math Phys 200(2):421–444, 1999) gave a uniform asymptotic estimate. Later, Zhang (Commun Math Phys 268(1):105–133, 2006) and Gluck (Nonlinear Anal 75(15):5787–5796, 2012) improved Li’s estimate in the sense of $$C^0$$ -convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao–Wang–Zhou (J Math Anal Appl 418:142–162, 2014) proved a similar asymptotic estimate of Li (1999). In this paper, we will prove a $$C^0$$ -convergence result in this boundary blow-up process. Our method is different from Gluck (2013), Zhang (2006).