Blowing-up solutions for a nonlocal Liouville type equation

Abstract

We consider the nonlocal Liouville type equation $$ (-Δ)^{\frac{1}{2}} u = \varepsilon κ(x) e^u, \quad u > 0, \quad \mbox{in } I, \qquad u = 0, \quad \mbox{in } \mathbb{R} \setminus I, $$ where $I$ is a union of $d\ge 2$ disjoint bounded intervals, $\kappa$ is a smooth bounded function with positive infimum and $\epsilon >0$ is a small parameter. For any integer $1\le m\le d$, we construct a family of solutions $(u_{\epsilon }{)}_{\epsilon }$ which blow up at $m$ interior distinct points of $I$ and for which $\epsilon {\int }_I\kappa e^{u_{\epsilon }}\to 2m\pi$, as $\epsilon \to 0$. Moreover, we show that, when $d=2$ and $m$ is suitably large, no such construction is possible.