Uniqueness of Positive Ground State Solutions of the Logarithmic Schrodinger Equation

We prove the uniqueness of positive ground state solutions of the problem , , and as . This equation is derived from the logarithmic Schrodinger equation , and also from the classical equation . For each , a positive ground state solution is . We combine with energy estimates and associated Ricatti equation estimates to prove that, for each , is the only positive ground state. We also investigate the stability of . Several open problems are stated.