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 $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1|u|^{p-2}u+\nu\alpha |u|^{\alpha-2}|v|^{\beta}u ~\hbox{in}~ \R^N,\\ -\Delta v+\lambda_2 v=\mu_2|v|^{q-2}v+\nu\beta |u|^{\alpha}|v|^{\beta-2}v ~\hbox{in}~ \R^N, \end{cases} $$ which arises in two-components Bose-Einstein condensates and involve attractive Sobolev subcritical or critical interactions, i. e., $\nu>0$ and $\alpha+\beta\leq 2^*$. This system is employed by seeking critical points of the associated variational functional with the constrained mass below $$\int_{\mathbb{R}^N}|u|^2 {\rm d}x=a, \quad \int_{\mathbb{R}^N}|v|^2 {\rm d}x=b.$$ In the mass mixed case, i. e., $2<p<2+\frac{4}{N}<q<2^*$, for some suitable $a,b,\nu$ and $\beta$, the system above admits two positive solutions. In particular, in the case $\alpha+\beta<2^*$, 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.