TITLE:
Normalized Solutions to a Planar Schrödinger-Poisson System with Ring-Shaped Potentials and Inhomogeneous Interactions
AUTHORS:
Xuanxuan Li
KEYWORDS:
Schrödinger-Poisson System, Normalized Solution, Logarithmic Convolution, Ringed-Shaped Potential
JOURNAL NAME:
Open Access Library Journal,
Vol.13 No.8,
August
25,
2026
ABSTRACT: This paper studies the constrained minimization problem for the Schrödinger-Poisson system with a ring-shaped potential and a logarithmic convolution term in
ℝ
2
. The energy functional is made well-defined by introducing a weighted Sobolev space
X
and decomposing the logarithmic kernel. Define the critical constant
a
*
=
| Q |
2
2
, where
Q
is the unique positive radially symmetric solution of the two-dimensional scalar field equation
−Δu+u=
u
3
. Under suitable assumptions on the interaction coefficient
K(
x
)
, the existence and nonexistence of minimizers for the minimization problem
e(
γ,a
)
are systematically studied. The main results show that minimizers exist when
a<
a
*
and do not exist when
a>
a
*
or
a=
a
*
(with
γ>0
). Furthermore, for
γ>0
, the asymptotic behavior of minimizers is characterized as
a↗
a
*
.