Ground states of nonlocal elliptic equations with general nonlinearities via Rayleigh quotient

2024/09/03 by Diego Ferraz, Ferraz, Diego, Edcarlos D. Silva +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2409.02205

openalex publication_date 2024/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is established ground states and multiplicity of solutions for a nonlocal Schrödinger equation (-Δ)^s u + V(x) u = λa(x) |u|^q-2u + b(x)f(u) in ℝ^N, u ∈ H^s(ℝ^N), where 00, under general conditions over the measurable functions a, b, V and f. The nonlinearity f is superlinear at infinity and at the origin, and does not satisfy any Ambrosetti-Rabinowitz type condition. It is considered that the weights a and b are not necessarily bounded and the potential V can change sign. We obtained a sharp λ^*> 0 which guarantees the existence of at least two nontrivial solutions for each λ∈ (0, λ^*). Our approach is variational in its nature and is based on the nonlinear Rayleigh quotient method together with some fine estimates. Compactness of the problem is also considered.

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