Uniqueness of bound states for sublinear elliptic equations

We investigate the uniqueness of radial bound state solutions to the sublinear elliptic equation $$\{\begin{array}{ll}-\mathrm{\Delta }u-u+\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u=0& \text{in }{\mathbb{R}}^n,\\ u(x)\to 0& \text{as }\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty },\end{array}$$ where $q\in (1,2)$ and $n\ge 2$. A distinctive feature of this problem is the non-Lipschitz singularity of the nonlinearity at the origin, which gives rise to compactly supported ground states and bound states. Using a shooting argument together with a detailed analysis of the linearized variation with respect to the initial value, we prove that for every prescribed integer $k\ge 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros, up to sign reflection and spatial translation. In addition, our analysis yields a classification of radial solutions according to the initial value and describes their behavior near the finite support boundary.