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Positive solutions to some systems of coupled nonlinear Schrödinger equations

We study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time-independent Schrödinger equations $$ - Δu_j+ λ_j u_j =μ_j u_j^3+ \sum_{k=1;k\neq j}^Nβ_{jk} u_ju_k^2,\quad u_j\in W^{1,2}(\mathbb{R}^n);\: j=1,\ldots,N $$ where $n=1,\,2,\, 3; \,λ_j,\, μ_j>0$ for $j=1,\ldots,N$, the coupling parameters $β_{jk}=β_{kj}\in \mathbb{R}$ 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.