Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

Abstract

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 \[ \begin{cases} -Δu =u-u^3-Λuv^2 -Δv =v-v^3-Λu^2v u,v > 0 \end{cases} \qquad \text{in ${\mathbb{R}}^N$, with $\Lambda >1$,} \] 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.