Super poly-harmonic property of solutions for Navier boundary problems on a half space

Authors

Wenxiong Chen, Yeshiva University
Yanqin Fang, University of WollongongFollow
Congming Li, Shanghai Jiaotong University

RIS ID

112126

Publication Details

Chen, W., Fang, Y. & Li, C. (2013). Super poly-harmonic property of solutions for Navier boundary problems on a half space. Journal of Functional Analysis, 265 (8), 1522-1555.

Abstract

In this paper, we consider the following poly-harmonic semi-linear equation with Navier boundary conditions on a half space R+n:(1){(-δ)mu=up,p>1,m≥1,u>0,inR+n,u=δu=⋯=δm-1u=0,on∂R+n. We first prove that the positive solutions of (1) are super poly-harmonic, i.e.(2)(-δ)iu>0,i=0,1,. .,m-1. Then, based on (2), we establish the equivalence between PDE (1) and the integral equation(3)u(x)=cnOU{ligature}R+n(1|x-y|n-2m-1|--y|n-2m)up(y)dy, where 1