| Contents |
| In this talk we consider solutions of the competitive elliptic system
\[
\begin{cases}
-\Delta u_i = - \sum_{j \neq i} u_i u_j^2 & \text{in $\mathbb{R}^N$} \\
u_i >0 & \text{in $\mathbb{R}^N$}
\end{cases} \qquad i=1,\dots,k,
\]
which appears in the analysis of phase separation phenomena for Bose-Einstein condensates with multiple states. The prototype of our main results is the following: for every $d >0$ there exists $h=h(d,N) \in \mathbb{N}$ such that if $(u_1,\dots,u_k)$ is a solution of the considered system and
\[
u_1(x)+\cdots+u_k(x) \le C(1+|x|^d) \qquad \text{for every $x \in \mathbb{R}^N$},
\]
then $k \le h(d,N)$. This means that a bound on the growth of a positive solution imposes a bound on the number of components $k$ of the solution itself. If $N=2$, the expression of $h(d,N)$ is explicit and optimal, while in higher dimension it can be characterized in terms of an optimal partition problem. We discuss the sharpness of our results and, as a further step, for every $N \ge 2$ we can prove the $1$-dimensional symmetry of the solutions satisfying suitable assumptions, extending known results which are available for $k=2$. The proofs rest upon a blow-down analysis and on some monotonicity formulae. |
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