Normalised solutions and limit profiles of the defocusing Gross-Pitaevskii-Poisson equation

TL;DR

This paper investigates the existence, shape, and asymptotic behavior of normalized solutions to the defocusing Gross-Pitaevskii-Poisson equation, revealing their convergence to known ground states depending on the Riesz potential parameter.

Contribution

The study provides new sharp asymptotic estimates for mass-energy curves and characterizes the limit profiles of solutions, depending on the Riesz potential parameter alpha.

Findings

Normalized solutions exist and form branches with specific mass-energy relations.

Solutions converge to ground states of the Choquard or Thomas-Fermi equations under rescaling.

The behavior of solutions varies critically with the parameter alpha.

Abstract

We study normalised solutions of the stationary Gross-Pitaevskii-Poisson (GPP) equation with a defocusing local nonlinear term, $$-\Delta u+\lambda u+|u|^2u =(I_\alpha*|u|^2)u\quad\text{in $\mathbb R^3$},\qquad\int_{\mathbb R^3}u^2dx=\rho^2,$$ where $\rho^2>0$ is the prescribed mass of the solutions, $\lambda\in\mathbb R$ is an a-priori unknown Lagrange multiplier, and $I_\alpha(x)=A_\alpha|x|^{3-\alpha}$ is the Riesz potential of order $\alpha\in(0,3)$. When $\alpha=2$ this problem appears in the models of self-gravitating Bose-Einstein condensates, which were proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We establish the existence of branches of normalised solutions to the GPP equation, paying special attention to the shape of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. The main novelty in this work is in the derivation of sharp asymptotic estimates on the mass-energy curves. These estimates allow us to show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation or to a compactly supported radial ground state of the integral Thomas-Fermi equation. The behaviour of normalized solutions depends sensitively on whether $\alpha$ is greater than, equal to, or less than one.