In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in arXiv:1609.03608 and est
ablish a quantization result for entire solutions of the singular n-Liouville equation.
We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in [10], extending Polyakovs formula. These corresp
ond to solutions of elliptic equations of Liouville type that are quasilinear, of mixed orders and of critical type. After studying existence, asymptotic behaviour and uniqueness of fundamental solutions, we prove a quantization property under blow-up, and then derive existence results via critical point theory.
We consider the two-dimensional mean field equation of the equilibrium turbulence with variable intensities and Dirichlet boundary condition on a pierced domain $$left{ begin{array}{ll} -Delta u=lambda_1dfrac{V_1 e^{u}}{ int_{Omega_{boldsymbolepsilon
}} V_1 e^{u} dx } - lambda_2tau dfrac{ V_2 e^{-tau u}}{ int_{Omega_{boldsymbolepsilon}}V_2 e^{ - tau u} dx}&text{in $Omega_{boldsymbolepsilon}=Omegasetminus displaystyle bigcup_{i=1}^m overline{B(xi_i,epsilon_i)}$} u=0 &text{on $partial Omega_{boldsymbolepsilon}$}, end{array} right. $$ where $B(xi_i,epsilon_i)$ is a ball centered at $xi_iinOmega$ with radius $epsilon_i$, $tau$ is a positive parameter and $V_1,V_2>0$ are smooth potentials. When $lambda_1>8pi m_1$ and $lambda_2 tau^2>8pi (m-m_1)$ with $m_1 in {0,1,dots,m}$, there exist radii $epsilon_1,dots,epsilon_m$ small enough such that the problem has a solution which blows-up positively and negatively at the points $xi_1,dots,xi_{m_1}$ and $xi_{m_1+1},dots,xi_{m}$, respectively, as the radii approach zero.
Entire solutions of the $n-$Laplace Liouville equation in $mathbb{R}^n$ with finite mass are completely classified.
