Abstract and Applied Analysis
Volume 2006 (2006), Article ID 89491, 22 pages

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

Imed Bachar

Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, Tunis 2092, Tunisia

Received 7 March 2005; Accepted 7 April 2005

Copyright © 2006 Imed Bachar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We establish a 3G-theorem for the iterated Green function of ()pm, (p1,m1), on the unit ball B of n(n1) with boundary conditions (/ν)j()kmu=0 on B, for 0kp1 and 0jm1. 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|)m1)=0, for 0kp1, 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).