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.

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 Wave equations associated with Liouville-type problems: global existence in time and blow-up criteria

2020 ◽  
Author(s):  
Weiwei Ao ◽  
Aleks Jevnikar ◽  
Wen Yang
Keyword(s):  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Mean Field ◽  
Liouville Type ◽  
Subcritical Case ◽  
Toda Systems ◽  
The Mean ◽  
Trudinger Inequality ◽  
Mean Field Equation

Abstract We are concerned with wave equations associated with some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated with the latter problems and second, to substantially refine the analysis initiated in Chanillo and Yung (Adv Math 235:187–207, 2013) concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the subcritical case and we give general blow-up criteria for the supercritical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser–Trudinger inequality.

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.

scholarly journals Mean field equations on a closed Riemannian surface with the action of an isometric group

2020 ◽  
Vol 31 (09) ◽  
pp. 2050072
Author(s):  
Yunyan Yang ◽  
Xiaobao Zhu
Keyword(s):  
Field Equation ◽  
Positive Integer ◽  
Existence Of Solutions ◽  
Mean Field ◽  
Riemannian Surface ◽  
Sufficient Condition ◽  
Field Equations ◽  
Mean Field Equations ◽  
The Mean ◽  
Mean Field Equation

Let [Formula: see text] be a closed Riemannian surface, [Formula: see text] be an isometric group acting on it. Denote a positive integer [Formula: see text], where [Formula: see text] is the number of all distinct points of the set [Formula: see text]. A sufficient condition for existence of solutions to the mean field equation [Formula: see text] is given. This recovers results of Ding–Jost–Li–Wang, Asian J. Math. (1997) 230–248 when [Formula: see text] or equivalently [Formula: see text], where Id is the identity map.

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
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