The Obstacle Problem at Zero for the Fractional p -Laplacian
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The Obstacle Problem at Zero for the Fractional p -Laplacian ´ Silvia Frassu1 · Eugenio M. Rocha2 · Vasile Staicu2 Received: 8 December 2019 / Accepted: 1 October 2020 / © Springer Nature B.V. 2020
Abstract In this paper we establish a multiplicity result for a class of unilateral, nonlinear, nonlocal problems with nonsmooth potential (variational-hemivariational inequalities), using the degree map of multivalued perturbations of fractional nonlinear operators of monotone type, the fact that the degree at a local minimizer of the corresponding Euler functional is equal one, and controlling the degree at small balls and at big balls. Keywords Obstacle problem · Fractional p-Laplacian · Operator of monotone type · Degree theory · Nonsmooth analysis Mathematics Subject Classification (2010) 47G20 · 47H05 · 47H11 · 49J40 · 49J52
1 Introduction Over the past few years, nonlocal operators have taken increasing importance, due to the fact that they appear in a number of applications, in such fields as game theory, finance, image processing, and optimization, see [2, 7, 9, 41] and the references therein. One of these operators is the fractional p-Laplacian, with p ∈ (1, ∞), a nonlinear and nonlocal operator, defined by |u(x) − u(y)|p−2 (u(x) − u(y)) dy, (1.1) (−)sp u(x) = 2 lim |x − y|N+ps →0+ RN \B (x) Vasile Staicu
[email protected] Silvia Frassu [email protected] Eug´enio M. Rocha [email protected] 1
Department of Mathematics and Computer Science, University of Cagliari, Via Ospedale 72, 09123, Cagliari, Italy
2
CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
S. Frassu et al.
for s ∈ (0, 1) and any sufficiently smooth function u : RN → R and all x ∈ RN . In the linear case, p = 2, the (1.1) gives the fractional Laplacian up to a dimensional constant C(N, p, s) > 0 (see [8, 19]). In [44] Teng studies hemivariational inequalities driven by nonlocal elliptic operator and he shows the existence of two nontrivial solutions, by applying critical point theory for nonsmooth functionals, while in [42] Servadei and Valdinoci prove Lewy-Stampacchia type estimates for variational inequalities driven by nonlocal operators. In [45] Xiang considers a variational inequality involving nonlocal elliptic operators, proving the existence of one solution, by exploiting variational methods combined with a penalization technique and Schauder’s fixed point theorem. In [1] Aizicovici, Papageorgiou and Staicu study the degree theory for the operator ∂C ϕ + ∂ψ, where ∂C ϕ is the Clarke generalized subdifferential of a locally Lipschitz functional ϕ, and ∂ψ is the subdifferential in the sense of convex analysis of a proper, convex and lower semicontinuous functional ψ. Their result regards the degree of an isolated minimizer for Euler functionals of the form ϕ + ψ and allows to study nonlinear variational inequalities with a nonsmooth potential function (variationalhemivariational inequalities). In the last decade hemivariational ineq
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