Blowing-up solutions for a nonlocal Liouville type equation

TL;DR

This paper constructs solutions to a nonlocal Liouville equation that blow up at multiple interior points as a small parameter tends to zero, revealing complex solution behavior and limitations in certain cases.

Contribution

It introduces a method to construct blow-up solutions for a nonlocal Liouville equation with multiple interior singularities, and identifies conditions where such solutions cannot exist.

Findings

Solutions blow up at specified interior points as epsilon approaches zero.

Existence of solutions depends on the number of intervals and blow-up points.

Nonexistence results are established for certain parameter regimes.

Abstract

We consider the nonlocal Liouville type equation $$ (-\Delta)^{\frac{1}{2}} u = \varepsilon \kappa(x) e^u, \quad u > 0, \quad \mbox{in } I, \qquad u = 0, \quad \mbox{in } \mathbb{R} \setminus I, $$ where $I$ is a union of $d \geq 2$ disjoint bounded intervals, $\kappa$ 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 $\varepsilon \int_I \kappa e^{u_\varepsilon} \, \rightarrow 2 m \pi$, as $\varepsilon \to 0$. Moreover, we show that, when $d = 2$ and $m$ is suitably large, no such construction is possible.