Qualitative analysis on logarithmic Schrödinger equation with general potential
In this paper, we study the existence, uniqueness, nondegeneracy and some qualitative properties of positive solutions for the logarithmic Schrödinger equations: \[ -Δu+ V(|x|) u=u\log u^2, u\in H^1(\mathbb R^N). \] 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 $V(r)=α_{1}\log r+α_2 r^{α_3}+α_4$ with $α_1>1-N$, $α_2, α_3\geq 0$ and $α_4\in\mathbb R$, which is repulsive for $α_1<0$ and $α_2=0$. We also investigate the connection between some power-law nonlinear Schrödinger equation with a critical frequency potential and the logarithmic-law Schrödinger equation with $V(r)=α\log r$, $α>1-N$, 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.