Uniqueness of positive solutions of double-power nonlinear stationary Schrödinger equations: the fixed frequency case and the fixed mass case

The existence of a positive radial solution to the problem \begin{align*} -Δu + λu = u^p + u^q, \quad u \in H^1(\mathbb{R}^N), ~~ N \ge 2, \end{align*} where $λ> 0$ and $1 < q < p < 2^* - 1$, has been known for a long time. For the pure-power nonlinearity, it is well known that this solution