Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

TL;DR

This paper proves the existence of infinitely many saddle-shaped positive solutions for certain elliptic systems with bistable nonlinearities, highlighting a stark contrast with cases where only constant solutions exist.

Contribution

It establishes the existence of infinitely many saddle-shaped solutions for a class of nonlinear elliptic systems with bistable nonlinearities in higher dimensions.

Findings

Infinitely many saddle-shaped solutions exist for the system when b3b1; > 1.

No non-constant solutions exist for b3b1; c9; (0,1].

The solutions exhibit saddle-shaped positivity in .

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