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

Pablo Figueroa

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arXiv · 2203.09731

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

Pablo Figueroa

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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.

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Pablo Figueroa. 2022-03-18. Bubbling solutions for mean field equations with variable intensities on compact Riemann surfaces. https://arxiv.org/abs/2203.09731

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