Blow-up behavior for a degenerate elliptic \sinh -Poisson equation with variable intensities

In this paper, we provide a complete blow-up picture for solution sequences to an elliptic sinh-Poisson equation with variable intensities arising in the context of the statistical mechanics description of two-dimensional turbulence, as initiated by Onsager. The vortex intensities are described in terms of a probability measure \(\mathcal P\) defined on the interval \([-1,1]\). Under Dirichlet boundary conditions we establish the exclusion of boundary blow-up points, we show that the concentration mass does not have residual \(L^1\)-terms (“residual vanishing”) and we determine the location of blow-up points in terms of Kirchhoff’s Hamiltonian. We allow \(\mathcal P\) to be a general Borel measure, which could be “degenerate” in the sense that \(\mathcal P(\{\alpha _-^*\})=0=\mathcal P(\{\alpha _+^*\})\), where \(\alpha _-^*=\min \mathrm {supp}\mathcal P\) and \(\alpha _+^*=\max \mathrm {supp}\mathcal P\). Our main results are new for the standard sinh-Poisson equation as well.