First curve of Fucik spectrum for the p-fractional Laplacian operator with nonlocal normal boundary conditions
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Date
2018-03-17
Authors
Goel, Divya
Goyal, Sarika
Sreenadh, Konijeti
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
In this article, we study the Fučik spectrum of the p-fractional Laplace operator with nonlocal normal derivative conditions which is defined as the set of all (α, b) ∈ ℝ2 such that
∧n,p (1 - α) (-∆)αpu + |u|p-2u = XΩε/ε (α(u+)p-1 -b(uˉ)p-1) in Ω,
Nα,pu = 0 in ℝn \ Ω̅,
has a non-trivial solution u, where Ω is a bounded domain in ℝn with Lipschitz boundary, p ≥ 2, n > pα, ε, α ∈ (0, 1) and Ωε ≔ {x ∈ Ω : d(x, ∂Ω) ≤ ε}. We show existence of the first non-trivial curve C of the Fučik spectrum which is used to obtain the variational characterization of a second eigenvalue of the problem defined above. We also discuss some properties of this curve C, e.g., Lipschitz continuous, strictly decreasing and asymptotic behavior and non-resonance with respect to the Fučik spectrum.
Description
Keywords
Nonlocal operator, Fucik spectrum, Steklov problem, Non-resonance
Citation
Goel, D., Goyal, S., & Sreenadh, K. (2018). First curve of Fucik spectrum for the p-fractional Laplacian operator with nonlocal normal boundary conditions. <i>Electronic Journal of Differential Equations, 2018</i>(74), pp. 1-21.
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Attribution 4.0 International