A Harnack-type inequality for a perturbed singular Liouville Equation

Abstract

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we obtain a Harnack-type inequality for sequences of solutions of the following perturbed Liouville equation, \begin{equation}\nonumber -\Delta v_n=\left({\epsilon_n^2+|x|^2}\right)^{\alpha_n}V_n(x)e^{\displaystyle v_n} \qquad\text{in} \,\,\, \Omega, \end{equation} where $\epsilon_n\to0^+$, $\alpha_n\to\alpha_\infty\in(-1,1)$, $\Omega$ is a bounded domain in $\mathbb{R}^2$ containing the origin and $V_n$ satisfies, \begin{equation}\nonumber 0<a\leq V_n\leq b<+\infty, \,\, V_n\in C^{0}(\Omega), \,\,V_n\to V \,\, \text{locally uniformly in}\,\,{\Omega}. \end{equation}