*Published Paper*

**Inserted:** 29 jul 2015

**Journal:** Calc. Var. Partial Differential Equations

**Volume:** 53

**Pages:** 689--718

**Year:** 2015

**Doi:** 10.1007/s00526-014-0764-3

**Links:**
link preliminary version

**Abstract:**

We study existence and phase separation, and the relation between these two aspects, of positive bound states for the nonlinear elliptic system \[ \begin{cases} - \Delta u_i + \lambda_i u_i = \sum_{j=1}^d \beta_{ij} u_j^2 u_i & \text{in $\Omega$} \\ u_1 =\cdots = u_d=0 & \text{on $\pa \Omega$}. \end{cases} \] This system arises when searching for solitary waves for the Gross-Pitaevskii equations. We focus on the case of simultaneous cooperation and competition , that is, we assume that there exist two pairs $(i_1,j_1)$ and $(i_2,j_2)$ such that $\beta_{i_1 j_1}>0$ and $\beta_{i_2 j_2}<0$. Our first main results establishes the existence of solutions with at least $m$ positive components for every $m \leq d$; any such solution is a minimizer of the energy functional $J$ restricted on a Nehari-type manifold $\mathcal{N}$. At a later stage, by means of level estimates on the constrained second differential of $J$ on $\mathcal{N}$, we show that, under some additional assumptions, any minimizer of $J$ on $\mathcal{N}$ has all nontrivial components. In order to prove this second result, we analyse the phase separation phenomena which involve solutions of the system in a not completely competitive framework.