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 $k > 2$.
Categories: math.AP