Existence, non-existence and asymptotic behaviour of blow-up entire solutions of quasi-linear elliptic equations

Abstract

Existence, non-existence and asymptotic behaviour of blow-up entire solutions of a class of quasi-linear elliptic equation div(|∇u| ^p−2∇u) = ρ(x)f(u), x ∈ R ^N is obtained. Under several hypotheses on the ρ(x) and f(r), we obtain the existence of blow-up entire solution. Existence and asymptotic behaviour is discussed by constructing lower and upper solution. For non-existence we explore radial symmetry, estimates on an associated integral equation and Keller–Osserman condition. The results of this article are new and extend previously known results.

Keywords:

• quasi-linear elliptic equation
• blow-up solution
• existence
• non-existence
• asymptotic behaviour

AMS Subject Classifications:

• 35J65
• 35B25

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant No. 10871060) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 08KJB110005).