Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

In this paper we prove the existence of infinitely many saddle-shaped positive solutions for non-cooperative nonlinear elliptic systems with bistable nonlinearities in the phase-separation regime. As an example, we prove that the system $$\begin{cases} -\Delta u =u-u^3-\Lambda uv^2 -\Delta v =v-v^3-\Lambda u^2v u,v > 0 \end{cases} \qquad \text{in $\mathbb{R}^N$, with $\Lambda>1$,}$$ has infinitely many saddle-shape solutions in dimension $2$ or higher. This is in sharp contrast with the case $\Lambda \in (0,1]$, for which, on the contrary, only constant solutions exist.