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Topological degree for mean field equations on \(S^2\). (English) Zbl 0964.35038
The very interesting paper under review is devoted to the topological degree theory for the mean field equation on the unit sphere \(S^2\) of \(\mathbb{R}^3\) equipped with the metric \(g_0\) induced from the flat metric of \(\mathbb{R}^3.\) Precisely, let \(f\) be a positive smooth function of \(S^2,\) \(\Delta\) the Laplace-Beltrami operator on \((S^2,g_0),\) \(d\mu\) the volume form with respect to \(g_0\) and \(\rho>0\) a constant. The mean field equation \[ \Delta \phi +\rho\left( {{f(y) e^\phi} \over {\int_{S^2} f(y) e^\phi d\mu}} - {1\over {4\pi}}\right)=0\quad \text{on} S^2 \tag{1} \] provides the Euler-Lagrange equation of the nonlinear functional \[ J_\rho(\phi)={{1}\over{2}} \int_{S^2} |\nabla \phi|^2 d\mu -\rho\log\left( \int_{S^2} f(y)e^\phi d\mu\right) \] for \(\phi\) satisfying the normalizing condition \[ \int_{S^2} \phi(y) d\mu(y)=0 \] and lying in the Sobolev space \(H^1(S^2)\) of functions with \(L^2\)-integrable first derivatives.
The author studies symmetric properties of the concentrated solutions to \((1)\) and apply them to the calculation of the Leray-Schauder degree in the cases \(f(y)\equiv 1\) and \(f(y)\equiv \exp (-\gamma\langle n, y\rangle)\) with positive constant \(\gamma\) and \(n\) being a unit vector in \(\mathbb{R}^3.\)

35J60 Nonlinear elliptic equations
58J05 Elliptic equations on manifolds, general theory
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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