Uniqueness of bound states to the logarithmic Schr\"odinger equation

This paper studies the uniqueness of bound states for the problem \Delta u + u\log u ^2=0, \quad u\in H^1(\RN), \quad n\geq 2, which arises from the logarithmic Schr\"odinger equation. We prove that for every integer $k\geq 1$, there exists a unique radial solution $u(r)=u(|x|)$ that has exactly $k$ simple zeros for $r>0$. This resolves an open problem posed by Troy [{Arch. Ration. Mech. Anal.} 222 (2016), 1581--1600] and confirms the Berestycki-Lions conjecture for the logarithmic nonlinearity. The proof combines the shooting method with suitable auxiliary functions introduced by Tang [{Invent. math.} 243 (2026), 245--291]. A major difficulty arises from the singular behavior of the nonlinearity $f(u)=u \log u^2$ at origin. We overcome it by establishing asymptotic convergence and sharp decay rates at infinity for any ground state or bound state. More precisely, every such solution satisfies \lim_{r\to\infty}\frac{ u'(r)}{u(r) \sqrt{\abs{ \log u^2(r)}}}=\lim_{r\to\infty}\frac{u'(r)}{ru(r)} = -1, \quad \limsup_{r\to\infty}|u(r)|e^{(\frac12-\epsilon)r^2}<\infty, ~~\forall \epsilon \in ( 0,\frac{1}{2}). These asymptotic behaviors are of independent interest and may be useful for other problems involving logarithmic nonlinearities.