Radial uniqueness and nondegeneracy for the Robin Lane-Emden equation

In this paper, we study positive radial solutions of $-\mathrm{\Delta }u=u^p$ in a ball $B_R\subset {\mathbb{R}}^N$, subject to the Robin boundary condition ${\mathrm{\partial }}_{\nu }u+\beta u=0$, where $N\ge 2$ and $\beta >0$. For every Sobolev subcritical exponent $p>1$, we prove the existence of a unique positive radial solution and its nondegeneracy with respect to radial perturbations. The proof relies on the strict monotonicity of the logarithmic slope of the normalized Lane-Emden profile, which follows from the radial Pohozaev identity. When $N\ge 3$, the same argument also recovers the classical critical existence range $0<\beta R<N-2$ and proves radial nondegeneracy throughout this interval. At $\beta R=(N-2)\mathrm{/}2$, we exhibit $N$ linearly independent nonradial solutions of the linearized problem.