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

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.

arXiv comment31 pages. arXiv admin note: substantial text overlap with arXiv:1210.6162