Positive radial solutions of p-Laplace equations on exterior domains

Abstract

This paper deals with the existence of positive radial solutions of the $ p $-Laplace equation $ \left\{\begin{array}{ll} -\Delta_p\,u= K(|x|)\,f(u)\,,\quad x\in\Omega\,,\qquad\qquad\\[6pt] \frac{\partial u}{\partial n}=0\,,\qquad x\in\partial\Omega,\\[6pt] \lim_{|x|\to\infty}u(x)=0\,, \end{array}\right. $ where $ \Omega = \{x\in {\mathbb{R}}^N:\, |x| > r_0\} $, $ N\ge 2 $, $ 1 < p < N $, $ K: [r_0, \, \infty)\to {\mathbb{R}}^+ $ is continuous and $ 0 < \int_{r_0}^{\infty}r^{N-1}K(r)\, dr < \infty $, $ f\in C({\mathbb{R}}^+, \, {\mathbb{R}}^+) $. Under the inequality conditions related to the asymptotic behaviour of $ f(u)/u^{p-1} $ at $ 0 $ and infinity, the existence results of positive radial solutions are obtained. The discussion is based on the fixed point index theory in cones.

Keywords

• p-laplace operator
• positive radial solution
• exterior domain
• cone
• fixed point index