Existence of positive solutions for some polyharmonic nonlinear boundary-value problems

Abstract

We present existence results for the polyharmonic nonlinear elliptic boundary-value problem (-Δ)m u = f(·,u) in B (∂/∂v)j u = 0 on ∂B, 0 ≤ j ≤ m - 1. (in the sense of distributions), where B is the unit ball in ℝn and n ≥ 2. The nonlinearity f(x,t) satisfies appropriate conditions related to a Kato class of functions Km,n. Our approach is based on estimates for the polyharmonic Green function with zero Dirichlet boundary conditions and on the Schauder fixed point theorem.