Positive solutions to some systems of coupled nonlinear Schr\"odinger equations

We study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time-independent Schr\"odinger equations $$- \Delta u_j+ \lambda_j u_j =\mu_j u_j^3+ \sum_{k=1;k\neq j}^N\beta_{jk} u_ju_k^2,\quad u_j\in W^{1,2}(\mathbb{R}^n);\: j=1,\ldots,N$$ where $n=1,\,2,\, 3; \,\lambda_j,\, \mu_j>0$ for $j=1,\ldots,N$, the coupling parameters $\beta_{jk}=\beta_{kj}\in \mathbb{R}$ for $j,k=1,\ldots,N$, $j\neq k$. Precisely, we prove the existence of nonnegative bound state solutions for suitable conditions on the coupling factors. Additionally, with more restrictive conditions on the coupled parameters, we show that the bound states founded are positive.