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

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 prescription of geodesic curvature on circle. build a family solutions which blow up, when hε approaches function h as ε → 0, at critical point harmonic extension provided some generic assumptions are satisfied.