Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation

Abstract

We consider solutions of the competitive elliptic system \[ \left\{ \begin{array}{ll} -Δu_i = - \sum_{j \neq i} u_i u_j^2 & \text{in ${\mathbb{R}}^N$} \\ u_i >0 & \text{in ${\mathbb{R}}^N$} \end{array}\right. \qquad i=1,\dots,k. \] 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 $\delta =\delta (k,N)\in \mathbb{N}$, increasing in $k$, such that if $(u_1,\dots ,u_k)$ is a solution and \[ u_1(x)+\cdots+u_k(x) \le C(1+|x|^d) \qquad \text{for every $x\in {\mathbb{R}}^N$}, \] 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 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.

27 pages

References in corpus (1)

• Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition

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