Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

2019/11/13 by Nicola Soave, Soave, Nicola
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1911.05602

arxiv created 2019/11/13 · arxiv updated 2019/11/14

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 \begincases -Δu =u-u³-Λuv² -Δv =v-v³-Λu²v u,v > 0 \endcases in ℝ^N, with Λ>1, has infinitely many saddle-shape solutions in dimension 2 or higher. This is in sharp contrast with the case Λ∈ (0,1], for which, on the contrary, only constant solutions exist.

Related

• Qualitative properties of saddle-shaped solutions to bistable diffusion equations
• Saddle-shaped solutions of bistable diffusion equations in all of ℝ^2m
• Positive solutions of elliptic systems with superlinear nonlinearities on the boundary
• Saddle-shaped solutions of bistable elliptic equations involving the half-Laplacian
• Existence of Strong and Nontrivial Solutions to Strongly Coupled Elliptic Systems