Infinitely many solutions for hemivariational inequalities involving the fractional Laplacian

Lijing Xi,Yuying Zhou

doi:10.1186/s13660-019-2253-x

Abstract

In the paper, we consider the following hemivariational inequality problem involving the fractional Laplacian:{(−Δ)su+λu∈α(x)∂F(x,u)x∈Ω,u=0x∈RN∖Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (-\\Delta )^{s}u+\\lambda u\\in \\alpha (x) \\partial F(x,u) & x \\in \\varOmega , \\\\ u=0 & x\\in \\mathbb{R} ^{N} \\backslash \\varOmega , \\end{cases} $$\\end{document} where Ω is a bounded smooth domain in mathbb{R} ^{N} with Ngeq 3, (-Delta )^{s} is the fractional Laplacian with sin (0,1), lambda >0 is a parameter, alpha (x): varOmega rightarrow mathbb{R} is a measurable function, F(x, u):varOmega times mathbb{R} rightarrow mathbb{R} is a nonsmooth potential, and partial F(x,u) is the generalized gradient of F(x, cdot ) at uin mathbb{R} . Under some appropriate assumptions, we obtain the existence of a nontrivial solution of this hemivariational inequality problem. Moreover, when F is autonomous, we obtain the existence of infinitely many solutions of this problem when the nonsmooth potentials F have suitable oscillating behavior in any neighborhood of the origin (respectively the infinity) and discuss the properties of the solutions.

Highlights

Boundary value problems involving fractional operators and more general nonlocal operators have attracted more interest since these operators appear in concrete applications in many fields, such as anomalous diffusion [1], quantum mechanics [2], obstacle problems [3], phase transition [4], minimal surface [5], and so on
Various papers deal with the existence and multiplicity of nontrivial solutions for the fractional Laplacian equations with superlinear or subcritical, critical, asymptotically linear nonlinearities, and some elliptic boundary problems involving the nonlocal integrodifferential operator are exploited, see for example [6,7,8,9,10] and the references therein
We note that the existence of infinitely many solutions for elliptic boundary value problems without the symmetric functionals is an important topic in nonlinear analysis, there are a lot of papers focused on the existence of infinitely many solutions of elliptic boundary value problems involving the local Laplacian and the p-Laplacian, see for example [11, 12]