Under consideration is the problem \[ -\text{div\,}(|\nabla u|^{p-2}\nabla u)=\lambda k(|x|)f(u)\;(x\in G=\{x:|x|>r_{0}>0\},\;p>1), \eqno{(1)} \]
\[ \;u|_{|x|=r_{0}}=0,\;\lim_{|x|\to \infty} u=0. \eqno{(2)} \] The functions \(f\in C[0,\infty)\cap C^{1}(0,\infty) \) and \(k\in C^{1}[r_{0},\infty)\) satisfy the conditions: \(f(0) < 0\), \(f'(s)> 0\) on \((0, +\infty)\) and \(f(s)\to \infty\) as \(s \to \infty\), \(\lim\sup_{s\to 0+}sf'(s) < \infty\), \(\lim_{s\to +\infty}f(s)/s^{p-1}=0\), \(k(r) > 0\) for \(r\geq r_{0}\) and \(k(r)\leq c/r^{n+\sigma}\) for some \(c>0\) and \(\sigma\in (0,(n-p)/(p-1))\), \(f(s)/s^{q}\) is nonincreasing on \([a,+\infty)\) for some \(a>0\) and \(q\in (0,p-1)\), the function \(r^{p(n-1)/(p-1)}k(r)\) is strictly increasing on \([r_{0},\infty)\). The main result of the article is that a positive radial solution to the problem (1), (2) is unique if the parameter \(\lambda \geq 1\) is sufficiently large.