TITLE:
Constraint Minimizers of the Gross-Pitaevskii Functional with Logarithmic Convolution and Ringed Shape Potential
AUTHORS:
Xuanxuan Li
KEYWORDS:
Gross-Pitaevskii Functional, Constraint Minimizers, Logarithmic Convolution, Ringed Shape Potential
JOURNAL NAME:
Open Access Library Journal,
Vol.13 No.2,
February
12,
2026
ABSTRACT: We consider a constrained variational problem where the energy functional includes a logarithmic convolution term and an external potential
(
| x |−A
)
2
. There is a threshold
a
*
∈(
0,∞
)
that we establish existence and nonexistence results for constraint minimizers: for
a<
a
*
, minimizers exist for any
γ≠0
; for
a>
a
*
with
γ≠0
and
a=
a
*
with
γ>0
, no minimizer exists. Furthermore, for
γ>0
and
a↗
a
*
, we analyze the limiting behavior of positive minimizers, showing that after suitable scaling, they converge to the standard ground state solution
Q(
x
)
of
−Δu+u=
u
3
in
ℝ
2
. We also derive asymptotic estimates for the location of the maximum points of minimizers.