Existence of positive radial solutions for the elliptic equations on an exterior domain
Volume 116 / 2016
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.
