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

Yongxiang Li¹ ; Huanhuan Zhang¹

¹ Department of Mathematics Northwest Normal University Lanzhou 730070, People's Republic of China

Annales Polonici Mathematici, Tome 116 (2016) no. 1, pp. 67-78

Résumé

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.

Détail

DOI : 10.4064/ap3633-12-2015

Keywords: discuss existence positive radial solutions semilinear elliptic equation begin cases delta hbox omega alpha beta tfrac partial partial hbox partial omega lim limits infty end cases where omega mathbb infty mathbb continuous int infty infty mathbb mathbb under conditions related asymptotic behaviour infinity existence positive radial solutions obtained conditions precise weaker superlinear sublinear growth conditions discussion based fixed point index theory cones