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Existence of positive radial solutions for the elliptic equations on an exterior domain

Volume 116 / 2016

Yongxiang Li, Huanhuan Zhang Annales Polonici Mathematici 116 (2016), 67-78 MSC: Primary 35J25; Secondary 47H11. DOI: 10.4064/ap3633-12-2015 Published online: 2 December 2015

Abstract

We discuss the existence of positive radial solutions of the semilinear elliptic equation $$ \begin{cases} -\Delta u = K(|x|) f(u),&\hbox{$x\in\Omega$,}\\ \alpha u+\beta \tfrac{\partial u}{\partial n}=0,&\hbox{$x\in\partial\Omega$,}\\ \lim\limits_{|x|\to\infty}u(x)=0, \end{cases} $$ where $\Omega=\{x\in \mathbb R^N:|x|>r_0\}$, $N\ge 3$, $K: [r_0, \infty)\to \mathbb R^+$ is continuous and $0<\int_{r_0}^{\infty}r K(r)\,dr<\infty$, $f\in C(\mathbb R^+, \mathbb R^+)$, $f(0)=0$. Under the conditions related to the asymptotic behaviour of $f(u)/u$ at $0$ and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the fixed point index theory in cones.

Authors

  • Yongxiang LiDepartment of Mathematics
    Northwest Normal University
    Lanzhou 730070, People's Republic of China
    e-mail
  • Huanhuan ZhangDepartment of Mathematics
    Northwest Normal University
    Lanzhou 730070, People's Republic of China
    e-mail

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