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

Abstract

We prove the existence of entire solutions with exponential growth for the semilinear elliptic system [\begin{cases} -Δu = -u v^2 & \text{in ${\mathbb{R}}^N$} -Δv= -u^2 v & \text{in ${\mathbb{R}}^N$} 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 $k>2$.

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