Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an