Blowing-up solutions for a nonlocal Liouville type equation

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 $d \geq 2$ disjoint bounded intervals, $κ$ is a smooth bounded function with positive infimum and $\varepsilon > 0$ is a small parameter. For any integer $1 \leq m \leq d$, we construct a family of solutions $(u_\varepsilon)_{\varepsilon}$ which blow up at $m$ interior distinct points of $I$ and for which $\varepsilon \int_I κe^{u_\varepsilon} \, \rightarrow 2 m π$, as $\varepsilon \to 0$. Moreover, we show that, when $d = 2$ and $m$ is suitably large, no such construction is possible.

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