School of Mathematics and Statistics
South-Central Minzu University
Wuhan, 430074, China
email: heyi19870113@163.com
Yi He
Abstract:
This article concerns the singularly of perturbed Gross-Pitaevskii equations in trapped
dipolar quantum gases,
$$\displaylines{
- \varepsilon^2 \Delta u + V(x)u +{\lambda _1}{| u |^2}u
+ {\lambda _2}(K * {| u |^2})u= 0 \quad \text{in } \mathbb{R}^3, \cr
u > 0,\quad u \in {H^1}({\mathbb{R}^3}),
}$$
where \(\varepsilon \) is a small positive parameter, \(\lambda_1, \lambda_2\in \mathbb{R}\),
\(*\) denotes the convolution, \( K(x) = \frac{{1 - 3{{\cos }^2}\theta }}{{{{| x |}^3}}}\)
and \(\theta = \theta(x)\) is the angle between the dipole axis determined by \((0,0,1)\)
and the vector \(x\). Moreover, the potential \(V\) satisfies
\(\liminf_{|x| \to \infty } V(x) > \inf_{\mathbb{R}^3} V(x) = 0\).
Under certain assumptions on \((\lambda_1, \lambda_2)\in \mathbb{R}^2\), we construct a family
of positive solutions \({u_\varepsilon } \in {H^1}({\mathbb{R}^3})\) whose
\({L^\infty }\) norm approaches \(0\) as \(\varepsilon \to 0\).
Our main results extend the results in Byeon and Wang [6]
which dealt with singularly perturbed Schrodinger equations with a local
nonlinearity, to the nonlocal Gross-Pitaevskii type equation.
Submitted January 23, 2026. Published July 2, 2026.
Math Subject Classifications: 35J20, 35J60.
Key Words: Dipolar quantum gases; standing waves; critical frequency.
DOI: 10.58997/ejde.2026.48
Show me the PDF file (411 KB), TEX file for this article.
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Yi He School of Mathematics and Statistics South-Central Minzu University Wuhan, 430074, China email: heyi19870113@163.com |
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