Multiple normalized solutions for two coupled Gross-Pitaevskii equations with attractive interactions and mass constriants

Abstract

We are concerned with the following system of two coupled time-independent Gross-Pitaevskii equations $$\{\begin{array}{l}-\hat{I}”u+\hat{I}{»}_{1}u=\hat{I}{¼}_1|u{|}^{p-2}u+\hat{I}½\hat{I}\pm |u{|}^{\hat{I}\pm -2}|v{|}^{2}uin{\mathbb{R}}^N,\\ -\hat{I}”v+\hat{I}{»}_{2}v=\hat{I}{¼}_2|v{|}^{q-2}v+\hat{I}½{\hat{I}}^2|u{|}^{\hat{I}}\pm |v{|}^{{\hat{I}}^2-2}vin{\mathbb{R}}^N,\end{array}$$ which arises in two-components Bose-Einstein condensates and involve attractive Sobolev subcritical or critical interactions, i. e., $\hat{I}½>0$ and $\hat{I}\pm +{\hat{I}}^2\le 2^{\ast }$. This system is employed by seeking critical points of the associated variational functional with the constrained mass below $${\int }_{{\mathbb{R}}^N}|u{|}^2\mathrm{d}x=a,{\int }_{{\mathbb{R}}^N}|v{|}^2\mathrm{d}x=b.$$ In the mass mixed case, i. e., $2<p<2+\frac{4}{N}<q<2^{\ast }$, for some suitable $a,b,\hat{I}½$ and ${\hat{I}}^2$, the system above admits two positive solutions. In particular, in the case $\hat{I}\pm +{\hat{I}}^2<2^{\ast }$, using variational methods on the $L^2$-ball, two positive solutions are obtained, one of which is a local minimizer and the second one is a mountain pass solution.

37 pages