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 $$\{\begin{array}{l}-\mathrm{\Delta }u=u-u^3-\mathrm{\Lambda }u{v}^2-\mathrm{\Delta }v=v-v^3-\mathrm{\Lambda }{u}^{2}vu,v>0\end{array}\text{in }{\mathbb{R}}^N\text{, with }\mathrm{\Lambda }>1\text{,}$$ has infinitely many saddle-shape solutions in dimension $2$ or higher. This is in sharp contrast with the case $\mathrm{\Lambda }\in (0,1]$, for which, on the contrary, only constant solutions exist.