Uniqueness of Bound States for Double-Power Scalar Field Equations

We study the uniqueness of bound states for the following scalar field equation $$ -Δu+u=-\varepsilon |u|^{q-1}u+|u|^{p-1}u \,\ \ \mbox{in}\,\ \mathbb R^N, \,\ \varepsilon>0,\tag*{$(*)$} $$ where $ N\ge3,\ 1 1$ satisfies $q \neq p$. For each integer $k \geq 1$, we prove that th