Concentration solutions to singularly prescribed Gaussian and geodesic curvatures problem

Abstract

We consider the following Liouville-type equation with exponential Neumann boundary condition: $$-\Delta \tilde{u}={\epsilon }^{2}K(x)e^{2\tilde{u}},x\in D,\frac{\partial \tilde{u}}{\partial n}+1=\epsilon \kappa (x)e^{\tilde{u}},x\in \partial D,$$ where $D\subset {\mathbb{R}}^2$ is the unit disc, ${\epsilon }^{2}K(x)$ and $\epsilon \kappa (x)$ 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.