Concentration solutions to singularly prescribed Gaussian and geodesic curvatures problem

TL;DR

This paper proves the existence of solutions that concentrate at specific points for a Liouville-type equation with exponential Neumann boundary conditions, involving prescribed Gaussian and geodesic curvatures on a disk.

Contribution

It establishes the first known existence results for concentration solutions in exponential Neumann boundary problems with prescribed curvatures.

Findings

Existence of concentration solutions under certain extremum conditions.

New results for exponential Neumann boundary problems.

Conditions involving the extremum of a combination of curvature functions.

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.