Concentration solutions to singularly prescribed Gaussian and geodesic curvatures problem

Abstract

We consider the following Liouville-type equation with exponential Neumann boundary condition: $$ -\Delta\tilde u = \varepsilon^2 K(x) e^{2\tilde u}, \quad x\in D, \qquad \frac{\partial \tilde u}{\partial n} + 1 = \varepsilon \kappa(x) e^{\tilde u}, \quad x\in\partial D, $$ where $D\subset \mathbb R^2$ is the unit disc, $\varepsilon^2 K(x)$ and $\varepsilon \kappa(x)$ stand for the prescribed Gaussian curvature and the prescribed geodesic curvature of the boundary, respectively. We prove the existence of concentration solutions if $\kappa(x) + \sqrt{K(x)+\kappa(x)^2}$ ($x\in\partial D$) has a strictly local extremum point, which is a total new result for exponential Neumann boundary problem.