Entire solutions with exponential growth for an elliptic system modeling phase-separation

TL;DR

This paper establishes the existence of entire solutions with exponential growth for a class of elliptic systems modeling phase separation, using approximation methods and monotonicity formulas, extending to systems with multiple components.

Contribution

It proves the existence of exponential growth solutions for elliptic systems and extends the construction to systems with more than two components.

Findings

Existence of entire solutions with exponential growth in $ ^N$

Construction method based on approximation and Almgren-type monotonicity

Extension of solutions to systems with more than two components

Abstract

We prove the existence of entire solutions with exponential growth for the semilinear elliptic system [\begin{cases} -\Delta u = -u v^2 & \text{in $\R^N$} -\Delta v= -u^2 v & \text{in $\R^N$} u,v>0, \end{cases}] for every $N \ge 2$. Our construction is based on an approximation procedure, whose convergence is ensured by suitable Almgren-type monotonicity formulae. The construction of \emph{some} solutions is extended to systems with $k$ components, for every $k > 2$.