On a two-component Bose-Einstein condensate with steep potential wells

TL;DR

This paper investigates ground state and multi-bump solutions of a two-component nonlinear Schrödinger system modeling Bose-Einstein condensates with steep, non-symmetric potentials, analyzing their concentration and phase separation behaviors as parameters vary.

Contribution

It establishes existence and multiplicity of solutions for large parameter values and explores their concentration and phase separation phenomena in non-radial settings.

Findings

Existence of ground state solutions for large mbda.

Multi-bump solutions with specific concentration behaviors.

Observation of phase separation in b^3.

Abstract

In this paper, we study the following two-component systems of nonlinear Schr\"odinger equations \begin{equation*} \left\{\aligned&\Delta u-(\lambda a(x)+a_0(x))u+\mu_1u^3+\beta v^2u=0\quad&\text{in }\bbr^3,\\ &\Delta v-(\lambda b(x)+b_0(x))v+\mu_2v^3+\beta u^2v=0\quad&\text{in }\bbr^3,\\ &u,v\in\h,\quad u,v>0\quad\text{in }\bbr^3,\endaligned\right. \end{equation*} where $\lambda,\mu_1,\mu_2>0$ and $\beta<0$ are parameters; $a(x), b(x)\geq0$ are steep potentials and $a_0(x),b_0(x)$ are sign-changing weight functions; $a(x)$, $b(x)$, $a_0(x)$ and $b_0(x)$ are not necessarily to be radial symmetric. By the variational method, we obtain a ground state solution and multi-bump solutions for such systems with $\lambda$ sufficiently large. The concentration behaviors of solutions as both $\lambda\to+\infty$ and $\beta\to-\infty$ are also considered. In particular, the phenomenon of phase separations is observed in the whole space $\bbr^3$. In the Hartree-Fock theory, this provides a theoretical enlightenment of phase separation in $\bbr^3$ for the 2-mixtures of Bose-Einstein condensates.