*Published Paper*

**Inserted:** 29 jul 2015

**Journal:** Advances in Math.

**Volume:** 279

**Pages:** 29--66

**Year:** 2015

**Doi:** 10.1016/j.aim.2015.03.015

**Links:**
link preliminary version

**Abstract:**

We consider solutions of the competitive elliptic system
\begin{equation}\tag{S}\label{syst 1}
\begin{cases}
-\Delta u_{i} = - \sum_{{j} \neq i} u_{i} u_{j}^{2} & \text{in $\R^N$} \\
u_{i} >0 & \text{in $\R^N$}
\end{cases} \qquad i=1,\dots,k.
\end{equation}
We are concerned with the classification of entire solutions, according with their growth rate. The prototype of our main results is the following: there exists a function $\delta=\delta(k,N) \in \N$, increasing in $k$, such that if $(u_1,\dots,u_k)$ is a solution of \eqref{syst 1} and
\[
u_1(x)+\cdots+u_k(x) \le C(1+

x

^d) \qquad \text{for every $x \in \R^N$},
\]
then $d \ge \delta$. This means that the number of components $k$ of the solution imposes an increasing in $k$ minimal growth on the solution itself. If $N=2$, the expression of $\delta$ 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 of \eqref{syst 1} 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.