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 ℝn(n≥2)(−△)mu+g(⋅,u)=0, in B (in the sense of distributions) limx→ξ∈∂B(u(x)/(1−|x|2)m−1)=0(ξ). 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.