On Uniqueness Conditions for Decreasing Solutions of Semilinear Elliptic Equations

Abstract

For $f\in C((0,\infty ))\cap C^1((0,\infty ))$ and $b>0$, existence and uniqueness of radial solutions $u=u(r)$ of the problem $\Delta u+f(u_{+})=0$ in ${\mathbb{R}}^n(n>2),u(0)=b$ and $u’(0)=0$ are well known. The uniqueness for the above problem with boundary conditions $u(R)=0$ and $u’(0)=0$ is less known beside the cases where $li{m}_{r\to \infty }u(r)=0$. It is our goal to give some sufficient conditions for the uniqueness of the solutions of the problem $D_{\alpha }u+f(u_{+})=0(r>0),u(p)=0$ and $u’(0)=0$ based only on the evolution of the functions $f(t)$ and $\frac{d}{dt}\frac{f(t)}{t}$.