Nonlocal elliptic systems via nonlinear Rayleigh quotient with general concave and coupling nonlinearities

2024/11/09 by Edcarlos D. Silva, Silva, Edcarlos D., Elaine Alves Leite +2
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2411.06169

openalex publication_date 2024/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of nonlocal elliptic systems: \(-Δ)^su +V₁(x)u = λ|u|^{p - 2}u+ \fracαα+βθ|u|^{α- 2}u|v|^β, in ℝ^N, (-Δ)^sv +V₂(x)v= λ|v|^{q - 2}v+ \fracβα+βθ|u|^α|v|^β-2v, in ℝ^N, (u, v) ∈ H^s(ℝ^N) × H^s(ℝ^N).. Here we mention that α> 1, β> 1, 1 ≤ p ≤ q < 2 < α+ β< 2^*_s, θ> 0, λ> 0, N > 2s, and s ∈ (0,1). Notice also that continuous potentials V₁, V₂: ℝ^N → ℝ satisfy some extra assumptions. Furthermore, we find the largest positive number λ^* > 0 such that our main problem admits at least two positive solutions for each λ∈ (0, λ^*). This can be done by using the nonlinear Rayleigh quotient together with the Nehari method. The main feature here is to minimize the energy functional in Nehari manifold which allows us to prove our main results without any restriction on size of parameter θ> 0.

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