Solutions to indefinite weakly coupled cooperative elliptic systems

2020/03/27 by Clapp, Mónica, Szulkin, Andrzej · 2 citations
#35B33 #35J50 #35J57 #58E30 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.12343

Abstract

We study the elliptic system \begincases -Δu₁ - κ₁u₁ = μ₁|u₁|^p-2u₁ + λα|u₁|^α-2|u₂|^βu₁,
-Δu₂ - κ₂u₂ = μ₂|u₂|^p-2u₂ + λβ|u₁|^α|u₂|^β-2u₂,
u₁,u₂∈ D^1,2₀(Ω), \endcases where Ω is a bounded domain in ℝ^N, N≥ 3, κ₁,κ₂∈ℝ, μ₁,μ₂,λ>0, α,β>1, and α+ β= p≤ 2^*:=(2N)/(N-2). For p∈ (2,2^*) we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for λ sufficiently large. If p=2^* and κ₁,κ₂>0 we establish the existence of a ground state for λ sufficiently large if, either N≥5, or N=4 and neither κ₁ nor κ₂ are Dirichlet eigenvalues of -Δ in Ω.

Cited by

• Quasilinear Elliptic Cooperative and Competitive Systems
• Least Energy Solutions for Cooperative and Competitive Schrödinger Systems with Neumann Boundary Conditions

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