Estimates for the Green function and singular solutions for polyharmonic nonlinear equation

Abstract

We establish a new form of the $3G$ theorem for polyharmonic Green function on the unit ball of ${ℝ}^n\left(n\ge 2\right)$ corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions $K_{m,n}$ containing properly the classical Kato class $K_n$. We exploit properties of functions belonging to $K_{m,n}$ to prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order $2m$.