A Harnack-type inequality for a perturbed singular Liouville Equation

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 -Δv_n=\left({ε_n^2+|x|^2}\right)^{α_n}V_n(x)e^{\displaystyle v_n} \qquad$\text{in}$ ,,, Ω, \end{equation} where ${\epsilon }_n\to 0^{+}$, ${\alpha }_n\to {\alpha }_{\mathrm{\infty }}\in (-1,1)$, $\mathrm{\Omega }$ 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}(Ω), ,,V_n\to V ,, $\text{locally uniformly in}$,,Ω. \end{equation}