Mathematics in Engineering, 2020, 2(3): 423-437. doi: 10.3934/mine.2020019 Research article Special Issues

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Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

Department of Mathematics, Politecnico di Milano, Via Bonardi 9, 20133, Milano, Italy

^†This contribution is part of the Special Issue: Qualitative Analysis and Spectral Theory for Partial Differential Equations
Guest Editor: Veronica Felli
Link: http://www.aimspress.com/newsinfo/1371.html

Received: , Accepted: , Published:

Special Issues: Qualitative Analysis and Spectral Theory for Partial Differential
Equations

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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.

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