Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 5 (2016), 1019-1041.

Multidimensional entire solutions for an elliptic system modelling phase separation

Nicola Soave and Alessandro Zilio

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For the system of semilinear elliptic equations

ΔV i = V i jiV j2,V i > 0  in 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 to higher dimensions N 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 systems and new sharp versions for monotonicity formulae of Alt–Caffarelli–Friedman type.

Article information

Anal. PDE, Volume 9, Number 5 (2016), 1019-1041.

Received: 16 July 2015
Revised: 5 February 2016
Accepted: 29 April 2016
First available in Project Euclid: 12 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B06: Symmetries, invariants, etc. 35B08: Entire solutions 35B53: Liouville theorems, Phragmén-Lindelöf theorems
Secondary: 35B40: Asymptotic behavior of solutions 35J47: Second-order elliptic systems

entire solutions of elliptic systems Liouville theorem nonlinear Schrödinger systems Almgren monotonicity formula optimal partition problems equivariant solutions


Soave, Nicola; Zilio, Alessandro. Multidimensional entire solutions for an elliptic system modelling phase separation. Anal. PDE 9 (2016), no. 5, 1019--1041. doi:10.2140/apde.2016.9.1019.

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