576 research outputs found
An Existence Result for the Mean Field Equation on Compact Surfaces in a Doubly Supercritical Regime
We consider a class of variational equations with exponential nonlinearities
on a compact Riemannian surface, describing the mean field equation of the
equilibrium turbulance with arbitrarily signed vortices. For the first time, we
consider the problem with both supercritical parameters and we give an
existence result by using variational methods. In doing this, we present a new
Moser-Trudinger type inequality under suitable conditions on the center of mass
and the scale of concentration of both e^u and e^{-u}, where u is the unknown
function in the equation.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1105.3701 by
other authors. The proof of Lemma 3.9 has been fixe
New existence results for the mean field equation on compact surfaces via degree theory
We consider a class of equations with exponential non-linearities on a
compact surface which arises as the mean field equation of the equilibrium
turbulence with arbitrarily signed vortices. We prove an existence result via
degree theory. This yields new existence results in case of a topological
sphere. The proof is carried out by considering the parity of the
Leray-Schauder degree associated to the problem. With this method we recover
also some known previous results
Analytic aspects of the Tzitz\'eica equation: blow-up analysis and existence results
We are concerned with the following class of equations with exponential
nonlinearities:
\Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2,
which is related to the Tzitz\'eica equation. Here are two smooth
positive functions. The purpose of the paper is to initiate the analytical
study of the above equation and to give a quite complete picture both for what
concerns the blow-up phenomena and the existence issue.
In the first part of the paper we provide a quantization of local blow-up
masses associated to a blowing-up sequence of solutions. Next we exclude the
presence of blow-up points on the boundary under the Dirichlet boundary
conditions.
In the second part of the paper we consider the Tzitz\'eica equation on
compact surfaces: we start by proving a sharp Moser-Trudinger inequality
related to this problem. Finally, we give a general existence result
A topological join construction and the Toda system on compact surfaces of arbitrary genus
We consider a Toda system of Liouville equations defined on a compact surface
which arises as a model for non-abelian Chern-Simons vortices. For the first
time the range of parameters , , is studied with a variational approach
on surfaces with arbitrary genus. We provide a general existence result by
means of a new improved Moser-Trudinger type inequality and introducing a
topological join construction in order to describe the interaction of the two
components
Blow-up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities
A class of equations with exponential nonlinearities on a compact Riemannian
surface is considered. More precisely, we study an asymmetric sinh-Gordon
problem arising as a mean field equation of the equilibrium turbulence of
vortices with variable intensities.
We start by performing a blow-up analysis in order to derive some information
on the local blow-up masses. As a consequence we get a compactness property in
a supercritical range.
We next introduce a variational argument based on improved Moser-Trudinger
inequalities which yields existence of solutions for any choice of the
underlying surface
A note on a multiplicity result for the mean field equation on compact surfaces
We are concerned with a Liouville-type equation with exponential
nonlinearities on a compact surface which describes the mean field equation of
the equilibrium turbulence with arbitrarily signed vortices. We provide the
first multiplicity result for this class of equations by using Morse theory
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