Abstract and Applied Analysis
Volume 2006 (2006), Article ID 89491, 22 pages
doi:10.1155/AAA/2006/89491

Estimates for the Green function and existence of positive solutions for higher-order elliptic equations

Abstract

We establish a 3G-theorem for the iterated Green function of (−∆)pm, (p≥1,m≥1), on the unit ball B of ℝn(n≥1) with boundary conditions (∂/∂ν)j(−∆)kmu=0 on ∂B, for 0≤k≤p−1 and 0≤j≤m−1. We exploit this result to study a class of potentials 𝒥m,n(p). Next, we aim at proving the existence of positive continuous solutions for the following polyharmonic nonlinear problems (−∆)pmu=h(‧,u), in D (in the sense of distributions), lim|x|→1((−∆)kmu(x)/(1−|x|)m−1)=0, for 0≤k≤p−1, where D=B or B\{0} and h is a Borel measurable function on D×(0,∞) satisfying some appropriate conditions related to 𝒥m,n(p).