Aleks Jevnikar

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 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 e u and e −u , where u is the unknown function in the equation.

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

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

\rho_1 \in (4k\pi , 4(k+1)\pi)

Symmetry and uniqueness of solutions to some Liouville-type equations and systems

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Non degeneracy, Mean Field Equations and the Onsager theory of 2D turbulence

k \in \mathbb{N}

A singular Sphere Covering Inequality: uniqueness and symmetry of solutions to singular Liouville-type equations

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Blow-up analysis and existence results in the supercritical case for an asymmetric mean field equation with variable intensities

\rho_2 \in (4\pi, 8\pi )

A general existence result for the Toda system on compact surfaces

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.

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

We are concerned with the following class of equations with exponential nonlinearities: