Uniqueness of bound states for sublinear elliptic equations

Abstract

We investigate the uniqueness of radial bound state solutions to the sublinear elliptic equation \[ \begin{cases} -Δu - u + |u|^{q-2}u = 0 & \text{in } \mathbb{R}^n,\cr u(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] 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.