Hardy-moser-trudinger Inequality And The Existence Of Weak Solutions Of Mean Field Equation

Our domain is Hardy Space in this paper. It uses the Hardy-Moser-Trudinger inequality to discuss a class of mean-field equations with Dirichlet boundary. The aim of this paper is to discuss the existence of weak solutions of the mean field equation.Let B1be an unit ball of R2. We use a symbol H to represent the Hardy Space. We discuss the mean field equation for a(x)=1/(1|x|2)2has singularity at point x=±1,(?)u∈H, ρ∈(-∞,16π).We discuss the existence of weak solutions of the mean field equation with Hardy-Moser-Trudinger inequality for ρ∈(-∞,8π), C>0.We discuss the existence of weak solutions of the mean field equation with the improved Hardy-Moser-Trudinger inequality for ρ∈(8π,16π), c>0.