scholarly journals On the existence and blow-up of solutions for a mean field equation with variable intensities

2016 ◽  
Vol 27 (4) ◽  
pp. 413-429 ◽  
Author(s):  
Tonia Ricciardi ◽  
Ryo Takahashi ◽  
Gabriella Zecca ◽  
Xiao Zhang
Keyword(s):  
Field Equation ◽  
Blow Up ◽  
Mean Field ◽  
Mean Field Equation

scholarly journals A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects

2018 ◽  
Vol 149 (2) ◽  
pp. 325-352 ◽  
Author(s):  
Aleks Jevnikar ◽  
Wen Yang
Keyword(s):  
Field Equation ◽  
Elliptic Problem ◽  
Blow Up ◽  
Existence Of Solutions ◽  
Mean Field ◽  
Probability Measures ◽  
Trudinger Inequality ◽  
Mean Field Equation

AbstractWe are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.

On the shape of blow-up solutions to a mean field equation

Nonlinearity ◽  
2006 ◽  
Vol 19 (3) ◽  
pp. 611-631 ◽  
Author(s):  
Daniele Bartolucci ◽  
Eugenio Montefusco
Keyword(s):  
Field Equation ◽  
Blow Up ◽  
Mean Field ◽  
Mean Field Equation

Non-axially symmetric solutions of a mean field equation on 𝕊2

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Changfeng Gui ◽  
Yeyao Hu
Keyword(s):  
Field Equation ◽  
Blow Up ◽  
Mean Field ◽  
Potential Energy Function ◽  
Great Circle ◽  
Regular Tetrahedron ◽  
Axially Symmetric ◽  
Platonic Solids ◽  
Equilateral Triangles ◽  
Mean Field Equation

Abstract We prove the existence of a family of blow-up solutions of a mean field equation on the sphere. The solutions blow up at four points where the minimum value of a potential energy function (involving the Green’s function) is attained. The four blow-up points form a regular tetrahedron. Moreover, the solutions we build have a group of symmetry {T_{d}} which is isomorphic to the symmetric group {S_{4}} . Other families of solutions can be similarly constructed with blow-up points at the vertices of equilateral triangles on a great circle or other inscribed platonic solids (cubes, octahedrons, icosahedrons and dodecahedrons). All of these solutions have the symmetries of the corresponding configuration, while they are non-axially symmetric.

Blow-up solutions for a mean field equation on a flat torus

2020 ◽  
Vol 69 (2) ◽  
pp. 453-485 ◽  
Author(s):  
Ze Cheng ◽  
Changfeng Gui ◽  
Yeyao Hu
Keyword(s):  
Field Equation ◽  
Blow Up ◽  
Mean Field ◽  
Flat Torus ◽  
Mean Field Equation

Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

2020 ◽  
Author(s):  
Guangze Gu ◽  
Changfeng Gui ◽  
Yeyao Hu ◽  
Qinfeng Li
Keyword(s):  
Field Equation ◽  
Level Sets ◽  
Mean Field ◽  
Flat Torus ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Flat Tori ◽  
The Mean ◽  
Symmetry Result ◽  
Mean Field Equation

Abstract We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

scholarly journals The nonlocal mean-field equation on an interval

2019 ◽  
Vol 22 (05) ◽  
pp. 1950028
Author(s):  
Azahara DelaTorre ◽  
Ali Hyder ◽  
Luca Martinazzi ◽  
Yannick Sire
Keyword(s):  
Field Equation ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mean Field ◽  
Higher Order ◽  
Dirichlet Boundary Conditions ◽  
Field Equations ◽  
Mean Field Equations ◽  
Blowing Up ◽  
Mean Field Equation

We consider the fractional mean-field equation on the interval [Formula: see text] [Formula: see text] subject to Dirichlet boundary conditions, and prove that existence holds if and only if [Formula: see text]. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of nonlocal type.

scholarly journals An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime

2013 ◽  
Vol 143 (5) ◽  
pp. 1021-1045 ◽  
Author(s):  
Aleks Jevnikar
Keyword(s):  
Field Equation ◽  
Existence Result ◽  
Mean Field ◽  
Type Inequality ◽  
Riemannian Surface ◽  
Supercritical Regime ◽  
Supercritical Parameters ◽  
The Mean ◽  
First Time ◽  
Mean Field Equation

We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean-field equation of the equilibrium turbulence 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 so, we present a new Moser–Trudinger-type inequality under suitable conditions on the centre of mass and the scale of concentration of both eu and e−u, where u is the unknown function in the equation.

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