Radial uniqueness and nondegeneracy for the Robin Lane-Emden equation

Abstract

In this paper, we study positive radial solutions of $-Δu=u^p$ in a ball $B_R\subset\mathbb{R}^N$, subject to the Robin boundary condition $\partial_νu+βu=0$, where $N\ge2$ and $β>0$. 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 $N\ge3$, the same argument also recovers the classical critical existence range $0<βR<N-2$ and proves radial nondegeneracy throughout this interval. At $βR=(N-2)/2$, we exhibit $N$ linearly independent nonradial solutions of the linearized problem.