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_j² & \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.