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 $\mathbb{R}^n (n\ge 2)(-\Delta)^m u + g(\cdot , u) =0$ $$, in $B$ (in the sense of distributions) $\lim_{x \to \zeta\in \partial B}(u(x) / (1- | x|^2 )^{m-1})= 0 {\zeta}$. Under appropriate conditions related to a Kato class on the nonlinearity $g (x,t)$, 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.