Standing waves with a critical frequency for the Gross-Pitaevskii equation in trapped dipolar quantum gases

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. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/48/abstr.html