Existence Results for Polyharmonic Boundary Value Problems in the Unit Ball

Abstract

Here we study the polyharmonic nonlinear elliptic boundary value problem on the unit ball $B$ in ${ℝ}^n\left(n\ge 2\right)$ ${\left(-△\right)}^{m}u+g\left(\cdot ,u\right)=0$, in $B$ (in the sense of distributions) ${lim}_{x\to \xi \in \partial B}(u\left(x\right)/{\left(1-{\left|x\right|}^2\right)}^{m-1})=0\left(\xi \right)$. Under appropriate conditions related to a Kato class on the nonlinearity $g\left(x,t\right)$, we give some existence results. Our approach is based on estimates for the polyharmonic Green function on $B$ with zero Dirichlet boundary conditions, including a 3G-theorem, which leeds to some useful properties on functions belonging to the Kato class.