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We study the existence of positive bound states for the nonlinear ellipticsystem \[ \begin{cases} - \Delta u_i + \lambda_i u_i = \sum_{j=1}^d \beta_{ij}u_j^2 u_i & \text{in $\Omega$} \\ u_1 =\cdots = u_d=0 & \text{on $\partial\Omega$}, \end{cases} \] where $d \ge 2$, $\beta_{ij}= \beta_{ji}$,$\beta_{ii},\lambda_i >0$, and $\Omega$ is either a bounded domain of$\mathbb{R}^N$, or $\Omega=\mathbb{R}^N$, with $N \le 3$. In light of itsapplicability in several physical contexts, the problem has been intensivelystudied in recent years, and several results concerning existence, multiplicityand qualitative properties of the solutions are available if either$\beta_{ij}\le 0$ for every $i \neq j$, or $\beta_{ij}>0$ for every $i \neq j$and some additional assumptions are satisfied. On the other hand, only verypartial results are known in the case of \emph{simultaneous cooperation andcompetition}, that is, when there exist two pairs $(i_1,j_1)$ and $(i_2,j_2)$such that $i_1 \neq j_1$, $i_2 \neq j_2$, $\beta_{i_1 j_1}>0$ and$\beta_{i_2,j_2}<0$. In this setting, we provide sufficient conditions on thecoupling parameters $\beta_{ij}$ in order to have a positive solution. Ourfirst main results establishes the existence of solutions with at least $m$positive components for every $m \le d$. Any such solution is a minimizer ofthe energy functional $J$ restricted on a \emph{Nehari-type manifold}$\mathcal{N}$. By means of level estimates on the constrained seconddifferential of $J$ on $\mathcal{N}$, we show that, under some additionalassumptions, any such minimizer has all nontrivial components. In order toprove this second result, we analyse the phase separation phenomena whichinvolve solutions of the system in a \emph{not completely competitive}framework.

On existence and phase separation of solitary waves for nonlinear Schr\"odinger systems modelling simultaneous cooperation and competition