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

Bachar, Imed

doi:https://doi.org/10.1155/AAA/2006/89491

Abstract and Applied Analysis

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Volume 2006 | Article ID 089491 | https://doi.org/10.1155/AAA/2006/89491

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

Imed Bachar¹

Received07 Mar 2005

Accepted07 Apr 2005

Published23 Feb 2006

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).

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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.

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