On the topological degree of the mean field equation with two parameters

Abstract

We consider the following class of equations with exponential nonlinearities on a compact surface M :which is associated to the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here h 1 , h 2 are smooth positive functions and ρ 1 , ρ 2 are two positive parameters.We start by proving a concentration phenomena for the above equation, which leads to a-priori bound for the solutions of this problem provided ρ i / ∈ 8πN, i = 1, 2. Then we study the blow up behavior when ρ 1 crosses 8π and ρ 2 / ∈ 8πN. By performing a suitable decomposition of the above equation and using the shadow system that was introduced for the SU (3) Toda system, we can compute the Leray-Schauder topological degree for ρ 1 ∈ (0, 8π) ∪ (8π, 16π) and ρ 2 / ∈ 8πN. As a byproduct our argument, we give new existence results when the underlying manifold is a sphere and a new proof for some known existence result.2000 Mathematics Subject Classification. 35J20, 35J60, 35R01. Key words and phrases. Geometric PDEs, Mean field equation, Topological degree. A.J. is partially supported by the PRIN project Variational and perturbative aspects of nonlinear differential problems.

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

“…Now, we consider the energy change from B δ k R k to B τ k . First we note that on ∂B δ k R k , by (25) and the latter estimate of the local energies we get…”

Blow-up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities

“…which has been very much studied recently [2,11,12,13,16,32]. For what concerns the quantization property (4) a similar result was derived in [15] (see also [17] for a similar approach).…”

Symmetry and uniqueness of solutions to some Liouville-type equations and systems

“…Using the first equation in (39), the definitions of w i in (29), and the fact that M = A + A ′ we get ∆u 1 + Ae u1 = Be u2 and hence…”

“…Equation (1) is related to the latter model when P is supported in two points. On the other hand, a deterministic assumption on the vortex intensities yields the following model (see [49]): Concerning the analysis of the latter equation we refer the interested readers to [1,23,24,25,26,28,29,30,31,45,47]. The arguments presented here do not apply to (3), and we postpone its analysis to a forthcoming paper.…”