New existence results for the mean field equation on compact surfaces via degree theory
Aleks Jevnikar
We consider the following class of equations with exponential nonlinearities on a closed surface $\Sg$: $$ - \D u = \rho_1 \left( \frac{h \,e^{u{\int_\Sg h \,e^{u \,dV_g - \frac{1{|\Sg| \right) - \rho_2 \left( \frac{h \,e^{-u{\int_\Sg h \,e^{-u \,dV_g - \frac{1{|\Sg| \right), $$ which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h$ is a smooth positive function and $\rho_1, \rho_2$ two positive parameters. By considering the parity of the Leray-Schauder degree associated to the problem, we prove solvability for $\rho_i \in (8\pi k, 8\pi(k+1)),\, k \in \N$. Our theorem provides a new existence result in the case when the underlying manifold is a sphere and gives a completely new proof for other known results.
1409.7798.tex
\documentclass[10pt]{amsart}


\usepackage{graphicx}

%\textwidth 16cm \oddsidemargin -0.01cm
%\evensidemargin -0.01cm
%\addtolength{\textheight}{3cm} \addtolength{\topmargin}{-2cm}



\usepackage[USenglish]{babel}
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
%\usepackage{natbib}
%\usepackage{showkeys}
\usepackage{esint}
\usepackage[all]{xy}
%\usepackage{hyperref}
\renewcommand{\a }{\alpha }
\renewcommand{\b }{\beta }
\renewcommand{\d}{\delta }
\newcommand{\D }{\Delta }
\newcommand{\Di }{\mathcal{D}^{1,2}(\R^n_+)}
\newcommand{\Dit }{\mathcal{D}^{1,2}(\R^3_+)}
\newcommand{\M}{\mathcal{M}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\e }{\varepsilon }
\newcommand{\g }{\gamma}
\renewcommand{\i }{\iota}
\newcommand{\G }{\Gamma }
\renewcommand{\l }{\lambda }
\renewcommand{\L }{\Lambda }
\newcommand{\m }{\mu }
\newcommand{\n }{\nabla }
\newcommand{\var }{\varphi }
\newcommand{\rh }{\rho }
\newcommand{\vrh }{\varrho }
\newcommand{\s }{\sigma }
\newcommand{\Sg }{\Sigma}
\newcommand{\Sig }{\Sigma}
\renewcommand{\t }{\tau }
\renewcommand{\th }{\theta }
\renewcommand{\o }{\omega }
\renewcommand{\O }{\Omega }
\newcommand{\oz }{{\ov{z}}}
\newcommand{\pa }{\partial}
\newcommand{\z }{\zeta }
\newcommand{\ov}{\overline}
\newcommand{\we}{\wedge}
\newcommand{\oo}{{\ov{1}}}
\newcommand{\wtilde }{\widetilde}
\newcommand{\wk}{\rightharpoonup}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newenvironment{pf}{\noindent{\sc Proof}.\enspace}{\rule{2mm}{2mm}\medskip}
\newenvironment{pfn}{\noindent{\sc Proof}}{\rule{2mm}{2mm}\medskip}
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newcommand{\R}{\mathbb{R}}
\renewcommand{\S}{\mathbb{S}}
\renewcommand{\P}{\mathbb{P}}
\renewcommand{\H}{\mathbb{H}}
\newcommand{\Rs}{\mbox{ \scriptsize I\hspace{-.15em}R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Zs}{\mbox{\scriptsize Z\hspace{-.25em}Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\no}{\noindent}
\newcommand{\dis}{\displaystyle}
\newcommand{\Sf}{\mathbb{S}}
\newcommand{\dotcup}{\ \ensuremath{\mathaccent\cdot\cup}\,}
\newcommand{\T}{\mathbb{T}}
\newcommand{\dkr}{{\bf d}}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}
\newtheorem{examples}[theorem]{Examples}
\newcommand{\bpr}{\begin{proposition}}
\newcommand{\epr}{\end{proposition}}
\newcommand{\bex}{\begin{example}\rm}
\newcommand{\eex}{\end{example}}

\def\bar#1{\mathcal B_{#1}}
\def\sj#1{\hbox{Sym}^{*{#1}}}
\def\sn{{\mathfrak S_n}}




\begin{document}


\newtheorem{lem}{Lemma}[section]
\newtheorem{pro}[lem]{Proposition}
\newtheorem{thm}[lem]{Theorem}
\newtheorem{rem}[lem]{Remark}
\newtheorem{cor}[lem]{Corollary}
\newtheorem{df}[lem]{Definition}





\title[The Mean Field equation on compact surfaces]{New existence results for the mean field equation on compact surfaces via degree theory}

\author {Aleks Jevnikar}

\address{SISSA, via Bonomea 265, 34136 Trieste (Italy).}


\email{ajevnika@sissa.it}

\keywords{Geometric PDEs, Leray-Schauder degree, Mean field equation.}

\subjclass[2000]{ 35J20, 35J61, 35R01.}





\begin{abstract}
We consider the following class of equations with exponential nonlinearities on a closed surface $\Sg$:
$$
  - \D u = \rho_1 \left( \frac{h \,e^{u}}{\int_\Sg
      h \,e^{u} \,dV_g} - \frac{1}{|\Sg|} \right) - \rho_2 \left( \frac{h \,e^{-u}}{\int_\Sg
      h \,e^{-u} \,dV_g} - \frac{1}{|\Sg|} \right),
$$
which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h$ is a smooth positive
function and $\rho_1, \rho_2$ two positive parameters. 
By considering the parity of the Leray-Schauder degree associated
to the problem, we prove solvability for $\rho_i \in (8\pi k, 8\pi(k+1)),\, k \in \N$. Our theorem provides a new
existence result in the case when the underlying manifold is a sphere and gives a completely new
proof for other known results.
\end{abstract}


\maketitle


\section{Introduction}

\no We are concerned with the equation
\begin{equation}
-\D u = \rho_1 \left(\frac{h\,e^u}{\int_\Sigma h\,e^u \,dV_g} - \frac{1}{|\Sigma|}\right) - \rho_2 \left(\frac{h\,e^{-u}}{\int_\Sigma h\,e^{-u} \,dV_g} - \frac{1}{|\Sigma|} \right) \hspace{0.3cm} \mbox{on $\Sigma$}, \label{eq}
\end{equation}
where $\D = \D_g$ is the Laplace-Beltrami operator, $\rho_1, \rho_2$ are two non-negative parameters, $h:\Sigma \rightarrow \mathbb{R}$ is a smooth positive function and $\Sigma$ is a compact orientable surface without boundary with Riemannian metric $g$ and total volume $|\Sigma|$. For the sake of simplicity, we will assume throughout this work that
$|\Sigma|=1$, which is no loss of generality thanks to a trivial rescaling argument�.

Problem \eqref{eq} plays an important role in mathematical physics as a mean field equation for the equilibrium turbulence with
arbitrarily signed vortices. It was first obtained by Joyce and Montgomery \cite{joy-mont} and  by Pointin and Lundgren \cite{point-lund} by means of different statistical arguments. Later, several authors adopted this model; we refer for example to \cite{cho}, \cite{lio}, \cite{mar-pul}, \cite{new} and the references therein. The case $\rho_1 = \rho_2$ is also related to the study of constant mean curvature surfaces, see \cite{wen1}, \cite{wen2}.

Before describing the main features of the problem and the known results, let us first consider the case $\rho_2 = 0$, namely the following standard Liouville-type equation:
\begin{equation} \label{eq2}
  - \D u = \rho \left( \frac{h \,e^{u}}{\int_\Sg
      h \,e^{u} \,dV_g} - 1 \right).
\end{equation}
Equation \eqref{eq2} appears in conformal geometry in the problem of finding a conformal metric for which the Gauss curvature is a prescribed function on $\Sg$, see \cite{bah-cor}, \cite{cha}, \cite{cha2}, \cite{li}, \cite{scho-zha}. Indeed, setting $\tilde g= e^{2v}g$, the Laplace-Beltrami operator for
the deformed metric is given by $\D_{\tilde g} = e^{-2v}\D_g$ and the change of the Gauss curvature is ruled by
$$
	-\D_g v = K_{\tilde g} e^{2v} - K_g,
$$  
where $K_g$ and $K_{\tilde g}$ are the Gauss curvatures of $(\Sg, g)$ and of $(\Sg, \tilde g)$ respectively.
Problem \eqref{eq2} also arises in mathematical physics as a mean field equation of Euler flows, see \cite{cal}, \cite{kies}. The literature on \eqref{eq2} is broad, and there are many results regarding existence, blow-up analysis, compactness of solutions, etc, see \cite{djlw}, \cite{djadli}, \cite{mal}, \cite{tar}.

\medskip

As many geometric problems, also \eqref{eq2} carries a lack of compactness, as its solutions might blow-up. It was proved in \cite{bre-mer}, \cite{li1} and \cite{li-shaf} that a quantization phenomenon occurs in this case. More precisely, taking a blow-up point $p$ for a sequence $(u_n)_n$ of solutions, we have
\begin{equation} \label{quant}
\lim_{r \to 0} \lim_{n \to + \infty}  \int_{B_r(p)} h \, e^{u_n} dV_g = 8 \pi.
\end{equation}
Roughly speaking, each blow-up point carries a quantized local mass. Moreover, the limit profile of solutions becomes close to a {\em bubble}, namely a function $U_{\lambda,p}$ defined as 
$$
	U_{\lambda,p}(y) = \log \left( \frac{4\lambda}{\bigr( 1 + \lambda \,d(p,y)^2 \bigr)^2} \right),
$$
where $y\!\in\!\Sg,\, d(p,y)$ stands for the geodesic distance and $\lambda$ is a large parameter. This limit function can be viewed as the logarithm of the conformal factor of the stereographic projection from $S^2$ onto $\R^2$, composed with a dilation.

Combining the local quantization \eqref{quant} with some further analysis, see for example \cite{bat-man}, \cite{bre-mer}, we have that the set of solutions
to \eqref{eq2} is uniformly bounded in $C^{2,\alpha}$, for any fixed $\alpha \in (0,1)$, provided $\rho \notin 8\pi\N$. It follows that one can define the Leray-Schauder degree associated to problem \eqref{eq2} with $\rho\in(8k\pi, 8(k+1)\pi), k\in\N$. In \cite{li1} it was shown that the degree is $1$ when $\rho < 8\pi$. By the homotopic invariance of the degree, it is easy to see that the same is independent of the function $h$, the metric of $\Sg$ and it is constant on each interval $(8k\pi, 8(k+1)\pi)$. In fact it depends only on $k\in\N$ and the topological structure of $\Sg$, as was proved in \cite{chen-lin}, where the authors provide the degree-counting formula
\begin{equation} \label{deg}
	\mbox{deg}(\rho) = \frac{1}{k!}(-\chi(\Sg) + 1) \cdots (-\chi(\Sg) + k),
\end{equation}
where $\chi(\Sg)$ denotes the Euler characteristic of $\Sg$. The proof of this result is carried out by analyzing the jump values of the degree after $\rho$ crossing the critical thresholds. Later, this result was rephrased in \cite{mal1} with a Morse theory point of view.

\medskip

On the other hand, in the general case when $\rho_2 \neq 0$, namely for equation \eqref{eq}, there are fewer results: for example the refined blow-up analysis of solutions of \eqref{eq} is not yet fully developed. Nevertheless, it was proven in \cite{jwyz} that the blow-up phenomenon yields a quantization property; for a blow-up point $p$ and a sequence $(u_n)_n$ of solutions relatively to $(\rho_{1,n}, \rho_{2,n})$ the authors obtained
$$
    \lim_{r \to 0} \lim_{n \to + \infty} \rho_{1,n} \frac{ \int_{B_r(p)} h \, e^{u_n} \,dV_g }{ \int_\Sg h \, e^{u_n} \,dV_g } \in 8 \pi \N, \qquad
    \lim_{r \to 0} \lim_{n \to + \infty} \rho_{2,n} \frac{ \int_{B_r(p)} h \, e^{-u_n} \,dV_g }{ \int_\Sg h \, e^{-u_n} \,dV_g } \in 8 \pi \N.
$$
Moreover, the case of positive multiples of $8 \pi$ may indeed occur, see \cite{es-we} and \cite{gr-pi}. The latter volume quantization implies that the set of solutions is compact for $(\rho_1, \rho_2) \notin (8 \pi \N \times \R) \cup (\R \cup 8 \pi \N)$. 

It follows that, as before, the degree associated to \eqref{eq} can still be defined outside this set of parameters. However, this strategy has not been yet investigated and the existence results mostly rely on a variational approach.

\medskip

Let us now briefly discuss the history of the problem and the previous results. First of all, the case $\rho_1, \rho_2 < 8\pi$ was considered in \cite{oh-su}. Ohtsuka and Suzuki proved that in this regime the associated energy functional is bounded
from below and coercive and therefore solutions can be found as global minima. If $\rho_i > 8\pi$ for some $i=1,2,$ then the problem becomes subtler as the functional is not bounded and a minimization technique is not possible any more. The first result in this direction is given in \cite{jwyz} for $\rho_1 \in (8\pi, 16\pi)$ and $\rho_2<8\pi$. Via a blow-up analysis Jost et al. proved the existence of solutions on a smooth, bounded, non-simply connected domain in $\R^2$ with homogeneous Dirichlet boundary condition. Later, in \cite{zhou}, Zhou generalized this result to any compact surface without boundary by reducing the problem to the one with $\rho_2 = 0$ and applying min-max theory. 

The next step was made in \cite{jev}, where for the first time a doubly supercritical regime is considered, namely $\rho_i \in (8\pi, 16\pi)$ for $i=1,2.$ The existence of solutions is proved using a min-max scheme based on a detailed description of the low sublevels of the energy functional. Finally, this was generalized in \cite{todatori} for a generic choice of the parameters $\rho_1 \in (8k\pi, 8(k+1)\pi)$ and $\rho_2 \in (8l\pi, 8(l+1)\pi),\, k,l \in \N$, under the assumption that $\Sg$ has positive genus.

   

\section{The main result}

We attack the problem with a different point of view and for the first time we analyze the associated Leray-Schauder degree. This is done in the spirit of \cite{mrd}, where a Toda system of Liouville equations arising from Chern-Simons theory was analyzed. More precisely, we study its parity and we observe that when both parameters stay in the same interval, i.e. $\rho_i \in (8k\pi, 8(k+1)\pi),\, k \in \N$ for $i=1,2$, the degree is always odd. The main result is the following.

\begin{thm}\label{teo}
Let $h>0$ be a smooth function and suppose \mbox{$\rho_i \in (8k\pi, 8(k+1)\pi)$,} $k \in \N$ for $i=1,2$. Then problem \eqref{eq} has a solution.
\end{thm}

Observe that we recover the result of \cite{jev} and some cases of \cite{todatori}: when $\Sg$ is homeomorphic to $S^2$ the above theorem yields a new existence result.

\

\begin{pf}
For some $\alpha \in (0,1)$ let $C^{2,\alpha}_0 (\Sg)$ be the class of $C^{2,\alpha}$ functions with zero average. Consider now the mapping $T : C^{2,\alpha}_0 (\Sg) \to C^{2,\alpha}_0 (\Sg)$ defined by
\begin{equation} \label{map}
	T(u) = (-\D)^{-1} \left( \rho_1 \left( \frac{h \,e^{u}}{\int_\Sg
      h \,e^{u} \,dV_g} - 1 \right) - \rho_2 \left( \frac{h \,e^{-u}}{\int_\Sg
      h \,e^{-u} \,dV_g} - 1 \right) \right),
\end{equation}
where $(-\D)^{-1}f, f\in C^\alpha(\Sg)$, is intended as the solution $v$, with zero average, of the problem $-\D v = f$, which is unique. We are concerned with the map $\Psi = Id - T$ and the solutions of equation \eqref{eq} will correspond to zeros of $\Psi$. 

Clearly, by elliptic regularity theory the operator $T$ is compact. Moreover, the set of the solutions is compact for parameters $(\rho_1, \rho_2) \notin (8 \pi \N \times \R) \cup (\R \times 8 \pi \N)$. Therefore, we can consider the associated degree deg$\left(\Psi_{(\rho_1,\rho_2)},B_r(0),0\right)$ which is well-defined for $r$ sufficiently large.

Consider now $\rho_i \in (8k\pi, 8(k+1)\pi),\, k \in \N$ for $i=1,2$. Letting $\rho = \frac 12 (\rho_1 + \rho_2)$, we perform the following homotopy which
takes place in a connected component of $\R^2 \setminus ((8 \pi \N \times \R) \cup (\R \times 8 \pi \N))$:
$$
	\mathfrak h(t) = (1-t)(\rho_1,\rho_2) + t (\rho,\rho).
$$
From the fact that the degree is constant along homotopies we obtain that
$$
	\mbox{deg}\left(\Psi_{(\rho_1,\rho_2)},B_r(0),0\right) = \mbox{deg}\left(\Psi_{(\rho,\rho)},B_r(0),0\right).
$$
Observe now that by the structure of $T$ we deduce
$$
	\Psi_{(\rho,\rho)}(-u) = -\Psi_{(\rho,\rho)}(u).
$$
Therefore, we conclude that $\Psi_{(\rho,\rho)}$ is an odd operator. By the Borsuk theorem, see \cite{kyner}, it follows that the associated degree is odd an hence non zero. This guarantees us the existence of a solution to equation \eqref{eq}.
\end{pf}


\

\begin{center}
\textbf{Acknowledgements}
\end{center}

\no The author would like to thank Professor Andrea Malchiodi for his kind help in preparing this paper.

The author is supported by the PRIN project \emph{Variational and perturbative aspects of nonlinear differential problems}.


\begin{thebibliography}{MMM}

\bibitem{bah-cor}
A. Bahri -- J.M. Coron, \emph{The scalar curvature problem on the standard three dimensional sphere}, J.
Funct. Anal., \textbf{95} (1991), pp. 106--172.

\bibitem{todatori}
L. Battaglia -- A. Jevnikar -- A. Malchiodi -- D. Ruiz, \emph{A general existence result for the Toda system on compact surfaces}, preprint (2013), arXiv http://arxiv.org/abs/1306.5404.

\bibitem{bat-man}
L. Battaglia -- G. Mancini, \emph{A note on compactness properties of singular Toda systems}, preprint (2014), arXiv http://arxiv.org/abs/1410.4991.

\bibitem{bre-mer}
H. Brezis -- F. Merle, \emph{Uniform estimates and blow-up behavior for solutions of $-\D u = V(x)\,e^u$ in two dimensions}, Comm. Partial Differential Equations, \textbf{16} (1991), pp. 1223--1253.

\bibitem{cal}
E. Caglioti -- P.L. Lions -- C. Marchioro -- M. Pulvirenti, \emph{A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description}, Part I, Comm. Math. Phys., \textbf{143} (1992), pp. 501--525. Part II, Comm. Math. Phys., \textbf{174} (1995), pp. 229--260.

\bibitem{cha}
S.Y.A. Chang -- M.J. Gursky -- P.C. Yang, \emph{The scalar curvature equation on 2- and 3- spheres}, Calc.
Var. and Partial Diff. Eq., \textbf{1} (1993), pp. 205--229.


\bibitem{cha2}
S.Y.A. Chang -- P.C. Yang, \emph{Prescribing Gaussian curvature on $S^2$}, Acta Math., \textbf{159} (1987), pp.
215--259.

\bibitem{chen-lin}
C.C. Chen -- C.S. Lin, \emph{Topological degree for a mean field equation on Riemann surfaces}, Comm. Pure
Appl. Math., \textbf{56} (2003), pp. 1667--1727.

\bibitem{cho}
A.J. Chorin, \emph{Vorticity and Turbulence}, Springer, New York, 1994.

\bibitem{djlw}
W. Ding -- J. Jost -- J. Li -- G. Wang, \emph{Existence results
for mean field equations}, Ann. Inst. Henri Poincar\'e, Anal. Non Lin{\`e}aire, \textbf{16} (1999), pp. 653--666.

\bibitem{djadli}
Z. Djadli, \emph{Existence result for the mean field problem on Riemann surfaces of all genus}, Comm. Contemp. Math., \textbf{10} (2008), pp. 205--220.

\bibitem{es-we}
P. Esposito -- J. Wei, \emph{Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation}, Calc. Var. Partial Differential Equations, \textbf{34} (2009), pp. 341--375.

\bibitem{gr-pi}
M. Grossi -- A. Pistoia, \emph{Multiple blow-up phenomena for the sinh-poisson equation}, Archive Rat. Mech. and Anal., \textbf{209} (2013), pp. 287--320.

\bibitem{jev}
A. Jevnikar, \emph{An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime}, Proc. Royal Soc. Edinburgh A, \textbf{143} (2013), pp. 1021--1045.

\bibitem{jwyz}
J. Jost -- G. Wang -- D. Ye -- C. Zhou, \emph{The blow up analysis of solutions of the elliptic sinh-Gordon equation}, Calc. Var. Partial Differential Equations, \textbf{31} (2008), pp. 263--276.

\bibitem{joy-mont}
G. Joyce -- D. Montgomery, \emph{Negative temperature states for the two-dimensional guiding-centre plasma}, J. Plasma Phys., \textbf{10} (1973), pp. 107--121.

\bibitem{kies}
M. Kiessling, \emph{Statistical mechanics of classical particles with logarithmic interactions}, Comm. Pure
Appl. Math., \textbf{46} (1993), pp. 27--56.

\bibitem{kyner}
W.T. Kyner, \emph{A Generalization of the Borsuk and Borsuk-Ulam Theorems}, Proc. Amer. Math. Soc., \textbf{7} (1956), pp. 1117--1119.

\bibitem{li1}
Y.Y. Li, \emph{Harnack type inequality: the method of moving planes}, Comm. Math. Phys., \textbf{200} (1999), pp. 421--444.

\bibitem{li}
Y.Y. Li, \emph{Prescribing scalar curvature on $S^n$ and related topics}, Part I, Journal of Differential Equations,
\textbf{120} (1995), pp. 319--410. Part II, \emph{Existence and compactness}, Comm. Pure Appl. Math., \textbf{49} (1996),
pp. 437--477.

\bibitem{li-shaf}
Y.Y. Li -- I. Shafrir, \emph{Blow-up analysis for solutions of $-\D u = V e^u$ in dimension two}, Indiana Univ. Math. J., \textbf{43} (1994), pp. 1255--1270.

\bibitem{lio}
P.L. Lions, \emph{On Euler Equations and Statistical Physics}, Scuola Normale Superiore, Pisa, 1997.

\bibitem{mal1}
A. Malchiodi, \emph{Morse theory and a scalar field equation on compact surfaces}, Adv. Differential Equations, \textbf{13} (2008), pp. 1109-�1129.

\bibitem{mal}
A. Malchiodi, \emph{Topological methods for an elliptic equation with exponential nonlinearities}, Discrete Contin. Dyn. Syst., \textbf{21} (2008), pp. 277--294.

\bibitem{mar-pul}
C. Marchioro -- M. Pulvirenti, \emph{Mathematical theory of incompressible nonviscous fluids}, Springer, New York, 1994.

\bibitem{mrd}
A. Malchiodi -- D. Ruiz, \emph{On the Leray-Schauder degree of the Toda system on compact surfaces}, preprint (2014), arXiv http://arxiv.org/abs/1311.7375.

\bibitem{new}
P.K. Newton, \emph{The N-Vortex Problem: Analytical Techniques}, Springer, New York, 2001.

\bibitem{oh-su}
H. Ohtsuka -- T. Suzuki, \emph{Mean field equation for the equilibrium turbulence and a related functional inequality}, Adv. Differential Equations, \textbf{11} (2006), pp. 281--304.

\bibitem{point-lund}
Y.B. Pointin -- T.S. Lundgren, \emph{Statistical mechanics of two-dimensional vortices in a bounded container}, Phys. Fluids, \textbf{19} (1976), pp. 1459--1470.

\bibitem{scho-zha}
R. Schoen -- D. Zhang, \emph{Prescribed scalar curvature on the n-sphere}, Calc. Var., \textbf{4} (1996), pp. 1--25.

\bibitem{tar}
G. Tarantello, \emph{Analytical, geometrical and topological aspects of a class of mean field equations on surfaces}, Discrete Contin. Dyn. Syst., \textbf{28}  (2010), pp. 931--973.

\bibitem{zhou}
C. Zhou, \emph{Existence of solution for mean-field equation for the equilibrium turbulence}, Nonlin. Analysis, \textbf{69} (2008), pp. 2541�-2552.

\bibitem{wen1}
H.C. Wente, \emph{Large solutions to the volume constrained Plateau problem}, Arch. Rational Mech.
Anal., \textbf{75} (1980/81), pp. 59--77. 

\bibitem{wen2}
H.C. Wente, \emph{Counterexample to a conjecture of H. Hopf}, Pacific J. Math., \textbf{121} (1986), pp. 193--243.

\end{thebibliography}

\end{document}
1409.7798.extracted_bib
{bah-cor A. Bahri -- J.M. Coron, The scalar curvature problem on the standard three dimensional sphere, J. Funct. Anal., 95 (1991), pp. 106--172.
1409.7798.extracted_bib
{todatori L. Battaglia -- A. Jevnikar -- A. Malchiodi -- D. Ruiz, A general existence result for the Toda system on compact surfaces, preprint (2013), arXiv http://arxiv.org/abs/1306.5404.
1409.7798.extracted_bib
{bat-man L. Battaglia -- G. Mancini, A note on compactness properties of singular Toda systems, preprint (2014), arXiv http://arxiv.org/abs/1410.4991.
1409.7798.extracted_bib
{bre-mer H. Brezis -- F. Merle, Uniform estimates and blow-up behavior for solutions of $-\D u = V(x)\,e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), pp. 1223--1253.
1409.7798.extracted_bib
{cal E. Caglioti -- P.L. Lions -- C. Marchioro -- M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Part I, Comm. Math. Phys., 143 (1992), pp. 501--525. Part II, Comm. Math. Phys., 174 (1995), pp. 229--260.
1409.7798.extracted_bib
{cha S.Y.A. Chang -- M.J. Gursky -- P.C. Yang, The scalar curvature equation on 2- and 3- spheres, Calc. Var. and Partial Diff. Eq., 1 (1993), pp. 205--229.
1409.7798.extracted_bib
{cha2 S.Y.A. Chang -- P.C. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math., 159 (1987), pp. 215--259.
1409.7798.extracted_bib
{chen-lin C.C. Chen -- C.S. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), pp. 1667--1727.
1409.7798.extracted_bib
{cho A.J. Chorin, Vorticity and Turbulence, Springer, New York, 1994.
1409.7798.extracted_bib
{djlw W. Ding -- J. Jost -- J. Li -- G. Wang, Existence results for mean field equations, Ann. Inst. Henri Poincar\'e, Anal. Non Lin{\`eaire, 16 (1999), pp. 653--666.
1409.7798.extracted_bib
{djadli Z. Djadli, Existence result for the mean field problem on Riemann surfaces of all genus, Comm. Contemp. Math., 10 (2008), pp. 205--220.
1409.7798.extracted_bib
{es-we P. Esposito -- J. Wei, Non-simple blow-up solutions for the Neumann two-dimensional sinh-Gordon equation, Calc. Var. Partial Differential Equations, 34 (2009), pp. 341--375.
1409.7798.extracted_bib
{gr-pi M. Grossi -- A. Pistoia, Multiple blow-up phenomena for the sinh-poisson equation, Archive Rat. Mech. and Anal., 209 (2013), pp. 287--320.
1409.7798.extracted_bib
{jev A. Jevnikar, An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime, Proc. Royal Soc. Edinburgh A, 143 (2013), pp. 1021--1045.
1409.7798.extracted_bib
{jwyz J. Jost -- G. Wang -- D. Ye -- C. Zhou, The blow up analysis of solutions of the elliptic sinh-Gordon equation, Calc. Var. Partial Differential Equations, 31 (2008), pp. 263--276.
1409.7798.extracted_bib
{joy-mont G. Joyce -- D. Montgomery, Negative temperature states for the two-dimensional guiding-centre plasma, J. Plasma Phys., 10 (1973), pp. 107--121.
1409.7798.extracted_bib
{kies M. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math., 46 (1993), pp. 27--56.
1409.7798.extracted_bib
{kyner W.T. Kyner, A Generalization of the Borsuk and Borsuk-Ulam Theorems, Proc. Amer. Math. Soc., 7 (1956), pp. 1117--1119.
1409.7798.extracted_bib
{li1 Y.Y. Li, Harnack type inequality: the method of moving planes, Comm. Math. Phys., 200 (1999), pp. 421--444.
1409.7798.extracted_bib
{li Y.Y. Li, Prescribing scalar curvature on $S^n$ and related topics, Part I, Journal of Differential Equations, 120 (1995), pp. 319--410. Part II, Existence and compactness, Comm. Pure Appl. Math., 49 (1996), pp. 437--477.
1409.7798.extracted_bib
{li-shaf Y.Y. Li -- I. Shafrir, Blow-up analysis for solutions of $-\D u = V e^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), pp. 1255--1270.
1409.7798.extracted_bib
{lio P.L. Lions, On Euler Equations and Statistical Physics, Scuola Normale Superiore, Pisa, 1997.
1409.7798.extracted_bib
{mal1 A. Malchiodi, Morse theory and a scalar field equation on compact surfaces, Adv. Differential Equations, 13 (2008), pp. 1109-�1129.
1409.7798.extracted_bib
{mal A. Malchiodi, Topological methods for an elliptic equation with exponential nonlinearities, Discrete Contin. Dyn. Syst., 21 (2008), pp. 277--294.
1409.7798.extracted_bib
{mar-pul C. Marchioro -- M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids, Springer, New York, 1994.
1409.7798.extracted_bib
{mrd A. Malchiodi -- D. Ruiz, On the Leray-Schauder degree of the Toda system on compact surfaces, preprint (2014), arXiv http://arxiv.org/abs/1311.7375.
1409.7798.extracted_bib
{new P.K. Newton, The N-Vortex Problem: Analytical Techniques, Springer, New York, 2001.
1409.7798.extracted_bib
{oh-su H. Ohtsuka -- T. Suzuki, Mean field equation for the equilibrium turbulence and a related functional inequality, Adv. Differential Equations, 11 (2006), pp. 281--304.
1409.7798.extracted_bib
{point-lund Y.B. Pointin -- T.S. Lundgren, Statistical mechanics of two-dimensional vortices in a bounded container, Phys. Fluids, 19 (1976), pp. 1459--1470.
1409.7798.extracted_bib
{scho-zha R. Schoen -- D. Zhang, Prescribed scalar curvature on the n-sphere, Calc. Var., 4 (1996), pp. 1--25.
1409.7798.extracted_bib
{tar G. Tarantello, Analytical, geometrical and topological aspects of a class of mean field equations on surfaces, Discrete Contin. Dyn. Syst., 28 (2010), pp. 931--973.
1409.7798.extracted_bib
{zhou C. Zhou, Existence of solution for mean-field equation for the equilibrium turbulence, Nonlin. Analysis, 69 (2008), pp. 2541�-2552.
1409.7798.extracted_bib
{wen1 H.C. Wente, Large solutions to the volume constrained Plateau problem, Arch. Rational Mech. Anal., 75 (1980/81), pp. 59--77.
1409.7798.extracted_bib
{wen2 H.C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math., 121 (1986), pp. 193--243.