Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

Nicola Soave; Nicola Soave

doi:10.3934/mine.2020019

Mathematics in Engineering

2020, Volume 2, Issue 3: 423-437. doi: 10.3934/mine.2020019

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Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

Nicola Soave ^,

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

Received: 12 November 2019 Accepted: 06 February 2020 Published: 28 February 2020

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 \gt 0 \end{cases} \qquad \text{in }\mathbb{R}^N, \text{with }\Lambda \gt 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.
- elliptic systems,
- entire solutions,
- saddle solutions,
- bistable nonlinearity,
- variational methods
Citation: Nicola Soave. Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity[J]. Mathematics in Engineering, 2020, 2(3): 423-437. doi: 10.3934/mine.2020019

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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 \gt 0 \end{cases} \qquad \text{in }\mathbb{R}^N, \text{with }\Lambda \gt 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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