Some existence results for variational inequalities with nonlocal fractional operators. (December 2019)

• Record Type: Journal Article
• Authors: Pimenta, Marcos Tadeu Oliveira; Servadei, Raffaella
• Is Part Of: Nonlinear analysis. Volume 189(2019)
• Publication Date: 2019-12
• Subjects: primary 35J87 49J40 -- secondary 47G20 45G05 35A15; Fractional Laplacian -- Variational inequalities -- Penalization method -- Variational methods; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.06.020
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we consider the following nonlocal fractional variational inequality u ∈ X 0 s ( Ω ), u ⩽ ψ a.e. in Ω, 〈 u, v − u 〉 X 0 s ( Ω ) − λ 〈 u, v − u 〉 2 ⩾ ∫ Ω f x, u ( x ), ( − Δ ) β u ( x ) ( v ( x ) − u ( x ) ) d x a a a a a a a a a a a a a a a a a a a a a a a a a a a a a for any v ∈ X 0 s ( Ω ), v ⩽ ψ a.e. in Ω, where Ω ⊂ R N is a smooth bounded open set with continuous boundary ∂ Ω, s ∈ ( 0, 1 ), N > 2 s, λ is a real parameter, f is function with subcritical growth, β ∈ ( 0, s ∕ 2 ) and ψ is the obstacle function. As it is well-known, the dependence of the nonlinearity f on the term ( − Δ ) β u makes non-variational the nature of this problem. Using an iterative technique and a penalization method, we get the existence of a nontrivial nonnegative solution for the problem under consideration, performing the Mountain Pass Theorem. This result can be seen as the extension of known existence theorem for variational inequalities driven by the Laplace operator (or more general uniformly elliptic operators) to the nonlocal fractional setting.