Abstract:We investigate the uniqueness of radial bound state solutions to the sublinear elliptic equation \[ \begin{cases} -\Delta u - u + |u|^{q-2}u = 0 & \text{in } \mathbb{R}^n, \\ u(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q\in(1,2)$ and $n\geq 2$. We prove that for any prescribed integer $k\geq 1$, the equation admits exactly one radial bound state solution with $k$ simple zeros. Furthermore, we consider the superlinear equation \[ \begin{cases} -\Delta u + u -|u|^{p-1}u = 0 & \text{in } \mathbb{R}^n, \\ u(x) \to 0 & \text{as } |x| \to \infty. \end{cases} \] While the uniqueness of radial bound state solutions for this equation was established by Tang (2026) for $n\geq 3$ and $1<p<\frac{n+2}{n-2}$, we provide the necessary arguments to show that this uniqueness result remains valid for the case $n=2$ with $p>1$.
From: Chengxiang Zhang [view email]
[v1]
Tue, 16 Jun 2026 12:10:07 UTC (41 KB)
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