Uniqueness of Positive Ground State Solutions of the Logarithmic Schrödinger Equation

Abstract

We prove the uniqueness of positive ground state solutions of the problem ${ {\frac {d^{2}u}{dr^{2}}} + {\frac {n-1}{r}}{\frac {du}{dr}} + u \ln(|u|) = 0}$, ${u(r) > 0~\forall r \ge 0}$, and ${(u(r),u'(r)) \to (0, 0)}$ as ${r \to \infty}$. This equation is derived from the logarithmic Schrödinger equation ${{\rm i}\psi_{t} = {\Delta} \psi + u \ln \left(|u|^{2}\right)}$, and also from the classical equation ${{\frac {\partial u}{\partial t}} = {\Delta} u +u \left(|u|^{p-1}\right) -u}$. For each ${n \ge 1}$, a positive ground state solution is ${ u_{0}(r) = \exp \left(-{\frac{r^2}{4}} + {\frac{n}{2}}\right),~0 \le r < \infty}$. We combine ${u_{0}(r)}$ with energy estimates and associated Ricatti equation estimates to prove that, for each ${n \in \left[1, 9 \right]}$, ${u_{0}(r)}$ is the only positive ground state. We also investigate the stability of ${u_{0}(r)}$. Several open problems are stated.