A blow-up phenomenon for a non-local Liouville-type equation

Abstract

We consider the non-local Liouville equation $${\left( { - \Delta } \right)^{{1 \over 2}}}u = {h_\varepsilon }{e^u} - 1\,\,\,\,\,{\rm{in}}\,\,{\mathbb{S}^1},$$ corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up, when hε approaches a function h as ε → 0, at a critical point of the harmonic extension of h provided some generic assumptions are satisfied.