Multidimensional entire solutions for an elliptic system modelling phase separation

Abstract

For the system of semilinear elliptic equations \[ ΔV_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in ${\mathbb{R}}^N$} \] we devise a new method to construct entire solutions. The method extends the existence results already available in the literature, which are concerned with the 2-dimensional case, also in higher dimensions $N\ge 3$. In particular, we provide an explicit relation between orthogonal symmetry subgroups, optimal partition problems of the sphere, the existence of solutions and their asymptotic growth. This is achieved by means of new asymptotic estimates for competing system and new sharp versions for monotonicity formulae of Alt-Caffarelli-Friedman type.

Final version: presentation of the results improved, and several minor corrections with respect to the first version

References in corpus (4)

• Uniform bounds for strongly competing systems: the optimal Lipschitz case
• Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation
• On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects
• On the uniqueness of solutions of an nonlocal elliptic system

Cited by in corpus (2)

• On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects
• On the Asymptotic Growth of Positive Solutions to a Nonlocal Elliptic Blow-up System Involving Strong Competition