Uniqueness of solutions to the mean field equations for the spherical Onsager vortex (Q1578961)

Let \(S^2\) be the unit sphere in \(\mathbb{R}^3\) and \(\langle n,y\rangle\) be the inner product of \(n\) and \(y\in S^2\) where \(n\) is a unit vector in \(\mathbb{R}^3\). In this paper, the author continues the work of a previous paper and studies the solution structure of the equation \[ \Delta_0\phi(y)+ {\exp(\alpha\phi(y)- \gamma\langle n,y\rangle)\over \int_{S^2} \exp(\alpha\phi(y)- \gamma\langle n,y\rangle d\mu}-{1\over 4\pi}= 0 \] on \(S^2\), where \(\Delta_0\) is the Beltrami-Laplace operator associated with the standard metric of \(S^2\), and \(\alpha\geq 0\) and \(\gamma\) are constants in \(\mathbb{R}\). This equation is the mean field equation arising from spherical Onsager vortex theory. The author studies axially symmetric solutions with respect to \(n\), satisfying \(\int_{S^2}\phi d\mu=0\). In particular, he proves the following results. (i) Let \(\gamma\geq 0\) and \(\alpha< 8\pi\). Then there exists a unique solution. If \(\gamma= 0\), then \(\phi\) is the trivial solution. If \(\gamma>0\), then \(\phi\) is axially symmetric with respect to \(n\). (iii) Let \(\gamma\geq 0\) and \(16\pi> \alpha\neq 4k(k+ 1)\pi\) for any integer \(k\geq 2\). Then there exist at least two axially symmetric solutions.