In this paper, we show the following equation
$\begin{cases} Δ u+u^{p}+λ u = 0&\text{ in }Ω,\\ u = 0&\text{ on }\partialΩ, \end{cases}$
has at most one positive radial solution for a certain range of $λ>0$. Here $p>1$ and $Ω$ is the annulus $\{x∈{{\mathbb{R}}^{n}}:a<|x|<b\}$, $0<a<b$. We also show this solution is radially non-degenerate via the bifurcation methods.
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