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

2022/03/18 by Figueroa, Pablo
#35B44 #35J15 #35J60 #35R01 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2203.09731

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 m₁ points and negatively at m-m₁ points with m≥ 1 and m₁∈\0,1,...,m\. Several examples in different situations illustrate our results in the sphere \mathbb S² and flat two-torus \mathbb T including non negative potentials with zero set non empty.

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