Superlinear fractional Φ-Laplacian type problems via the nonlinear Rayleigh quotient with two parameters

2025/07/21 by de Assis, L. R. S., Carvalho, M. L. M., Silva, Edcarlos D. +1
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.15514

Abstract

In this work, we establish the existence and multiplicity of weak solutions for nonlocal elliptic problems driven by the fractional Φ-Laplacian operator, in the presence of a sign-indefinite nonlinearity. More specifically, we investigate the following nonlocal elliptic problem: \(-Δ_Φ)^s u +V(x)u · amp; = · amp; μa(x)|u|^q-2u-λ|u|^p-2u in ℝ^N,
u∈ W^s,Φ(ℝ^N), · amp; · amp; . where s ∈ (0,1), N ≥ 2 and μ, λ>0. Here, the potentials V, a : ℝ^N → ℝ satisfy some suitable hypotheses. Our main objective is to determine sharp values for the parameters λ> 0 and μ> 0 where the Nehari method can be effectively applied. To achieve this, we utilize the nonlinear Rayleigh quotient along with a detailed analysis of the fibering maps associated with the energy functional. Additionally, we study the asymptotic behavior of the weak solutions to the main problem as λ→ 0 or μ→ +∞.

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• Nontrivial solutions for nonlinear problems driven by a superposition of fractional p-Laplacians with Neumann boundary conditions