Quasilinear elliptic problems via nonlinear Rayleigh quotient

2024/10/01 by Edcarlos D. Silva, Marcos L. M. Carvalho, Silva, Edcarlos D. +5 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2410.00861

openalex publication_date 2024/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems \ -Δ_Φu = λa(x) |u|^q-2u + |u|^p-2u, · amp; x∈Ω, u = 0, · amp; x ∈ ∂ Ω, . where Ω⊂ ℝ^N, N ≥ 2, is a smooth bounded domain, 1 < q < ℓ ≤ m < p < ℓ^* and Φ: ℝ → ℝ is suitable N-function. The main feature here is to show whether the Nehari method can be applied to find the largest positive number λ^* > 0 in such way that our main problem admits at least two distinct solutions for each λ∈ (0, λ^*). Furthermore, using some fine estimates and some extra assumptions on Φ, we prove the existence of at least two positive solutions for λ= λ^* and λ∈ (λ^*, λ) where λ > λ^*.

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