Abstract
Abstract We are concerned with the class of equations with exponential nonlinearities - Δ u = ρ 1 ( h e u ∫ Σ h e u 𝑑 V g - 1 | Σ | ) - ρ 2 ( h e - u ∫ Σ h e - u 𝑑 V g - 1 | Σ | ) $-\Delta u=\rho_{1}\Biggl{(}\frac{he^{u}}{\int_{\Sigma}he^{u}\,dV_{g}}-\frac{1}% {|\Sigma|}\Biggr{)}-\rho_{2}\Biggl{(}\frac{he^{-u}}{\int_{\Sigma}he^{-u}\,dV_{% g}}-\frac{1}{|\Sigma|}\Biggr{)}$ on a compact surface Σ, which describes the mean field equation of equilibrium turbulence with arbitrarily signed vortices. Here, h is a smooth positive function and ρ 1 , ρ 2 ${\rho_{1},\rho_{2}}$ are two positive parameters. We provide the first multiplicity result for this class of equations by using Morse theory.
Highlights
In this paper we consider the following mean field equation (1)on Σ, where ∆ = ∆g is the Laplace-Beltrami operator, ρ1, ρ2 are two non-negative parameters, h : Σ → R is a smooth positive function and Σ is a compact orientable surface without boundary with Riemannian metric g and total volume |Σ|
We provide the first multiplicity result for this class of equations by using
Several authors worked on this model; we refer for example to [13, 28, 32, 35, 36] and the references therein
Summary
Another motivation for the study of (4) is in mathematical physics as it models the mean field equation of Euler flows, see [7, 24] This problem has been studied widely by lots of authors and there are many results regarding existence, blow-up analysis, compactness of solutions, etc, see [15, 16, 31, 42]. For larger values of the parameter, the first step was done in [11], where an improved Moser-Trudinger is presented; roughly speaking, the more the function eu is spread over the surface Σ, the better is the constant in the inequality and, as a consequence, one gets new lower bounds on the functional (7) Improving this result, in [16] and [31] the authors were able to deduce a general existence result.
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