Qualitative analysis on logarithmic Schr\"odinger equation with general potential

In this paper, we study the existence, uniqueness, nondegeneracy and some qualitative properties of positive solutions for the logarithmic Schr\"odinger equations: $$-\Delta u+ V(|x|) u=u\log u^2, u\in H^1(\mathbb R^N).$$ Here $N\geq 2$ and $V\in C^2((0,+\infty))$ is allowed to be singular at $0$ and repulsive at infinity (i.e., $V(r)\to-\infty$ as ${r\to\infty}$). Under some general assumptions, we show the existence, uniqueness and nondegeneracy of this equation in the radial setting.Specifically, these results apply to singular potentials such as $V(r)=\alpha_{1}\log r+\alpha_2 r^{\alpha_3}+\alpha_4$ with $\alpha_1>1-N$, $\alpha_2, \alpha_3\geq 0$ and $\alpha_4\in\mathbb R$, which is repulsive for $\alpha_1<0$ and $\alpha_2=0$. We also investigate the connection between some power-law nonlinear Schr\"odinger equation with a critical frequency potential and the logarithmic-law Schr\"odinger equation with $V(r)=\alpha\log r$, $\alpha>1-N$, proving convergence of the unique positive radial solution from the power type problem to the logarithmic type problem. Under a further assumption, we also derive the uniqueness and nondegeneracy results in $H^1(\mathbb R^N)$ by showing the radial symmetry of solutions.