Uniqueness of least energy solutions of the fractional Lane-Emden equation in the ball

We prove uniqueness of least-energy solutions for a class of semilinear equations driven by the fractional laplacian, under homogeneous Dirichlet exterior conditions, when the underlying domain is a ball $B \subset \mathbb{R}^N$. The nonlinearity is of the form $$ f(u) = \lambda u + u^p,$$ where $p$ is superlinear and subcritical, and $\lambda < \lambda_1(B)$, where $\lambda_1(B)$ is the first eigenvalue of the fractional laplacian in $B$. For $\lambda= 0$, we recover the fractional Lane-Emden equation. The proof makes use of Morse theory, and is inspired by some results obtained by C. S. Lin in the '90s