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Uniqueness of solutions to the mean field equations for the spherical Onsager vortex. (English) Zbl 0968.35045
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.

MSC:
35J60 Nonlinear elliptic equations
35Q75 PDEs in connection with relativity and gravitational theory
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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