Some existence results for variational inequalities with nonlocal fractional operators

Abstract

In this paper we consider the following nonlocal fractional variational inequality u∈X0 s(Ω),u⩽ψa.e. in Ω,〈u,v−u〉X0 s(Ω)−λ〈u,v−u〉2⩾∫Ωfx,u(x),(−Δ)βu(x)(v(x)−u(x))dxaaaaaaaaaaaaaaaaaaaaaaaaaaaaafor anyv∈X0 s(Ω),v⩽ψa.e. in Ω, where Ω⊂RN is a smooth bounded open set with continuous boundary ∂Ω, s∈(0,1), N>2s, 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)βu 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.