Plenary talk

# Blowing-up solutions for a nonlocal Liouville type equation

We consider the nonlocal Liouville type equation

$$ (-\Delta)^{\frac12}u = \epsilon \kappa(x) e^{u}, u > 0, \text{ in } I, \quad u = 0, \text{ in } \mathbb{R} \setminus I, $$

where $I$ is a union of $d \geq 2$ disjoint bounded intervals, $\kappa$ is a smooth bounded function with positive infimum and $\epsilon > 0$ is a small parameter. For any integer $1 \leq m \leq d$, we construct a family of solution $(u_{\epsilon})_ \epsilon$ which blow-up at $m$ distinct points of $I$ and for which $\epsilon \int_{I} \kappa(x) e^{u} \to 2m\pi$ as $\epsilon \to 0$. Moreover, we show that, when $d = 2$ and $m$ is suitably large, no such construction is possible. The talk is based on a joint work with Matteo Cozzi (Milano, Italy).