Global structure of the solution set for a semilinear elliptic problem related to the Liouville Equation on an annulus

Abstract

A semilinear elliptic problem related to the Liouville equation on a two-dimensional annulus is studied. The problem appears as the limiting problem of the Liouville equation as the inside radius of the annulus tends to 0, and is derived by the method of matched asymptotic expansions. Our concern is the solution set of the problem in the bifurcation diagram. We find explicit solutions including non-radially symmetric solutions and determine the connected component containing the solutions. As a consequence, we provide a suggestive evidence for the global structure of the solution set of the Liouville equation.