Bubbling solutions for mean field equations with variable intensities on compact Riemann surfaces

TL;DR

This paper proves the existence of bubbling solutions for a mean field equation related to turbulence on compact Riemann surfaces, using Lyapunov-Schmidt reduction, with examples on spheres and tori.

Contribution

It establishes sufficient conditions for bubbling solutions with variable intensities on Riemann surfaces, extending previous results to asymmetric sinh-Poisson problems.

Findings

Existence of bubbling solutions blowing up at multiple points.

Solutions can be positive or negative at different points.

Applications to spheres and tori with specific potential conditions.

Abstract

For an asymmetric sinh-Poisson problem arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of bubbling solutions on compact Riemann surfaces. By using a Lyapunov-Schmidt reduction, we find sufficient conditions under which there exist bubbling solutions blowing up at $m$ different points of $S$: positively at $m_1$ points and negatively at $m-m_1$ points with $m\ge 1$ and $m_1\in\{0,1,...,m\}$. Several examples in different situations illustrate our results in the sphere $\mathbb S^2$ and flat two-torus $\mathbb T$ including non negative potentials with zero set non empty.