2021/09/30 by Zhang, Chengxiang, Zhang, Luyu
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2109.14858
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=ulog u2, u∈ H1(\mathbb RN). Here N≥ 2 and V∈ C2((0,+∞)) is allowed to be singular at 0 and repulsive at infinity (i.e., V(r)→-∞ as r→∞). 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)=α1log r+α2 rα3+α4 with α1>1-N, α2, α3≥ 0 and α4∈\mathbb R, which is repulsive for α1<0 and α2=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)=αlog 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 H1(\mathbb RN) by showing the radial symmetry of solutions.