Infinitely many solutions for hemivariational inequalities involving the fractional Laplacian

In the paper, we consider the following hemivariational inequality problem involving the fractional Laplacian: integral(-Delta)(s)u + lambda u is an element of alpha(x)partial derivative F(x,u) x is an element of Omega, u = 0 x is an element of R-N\Omega, where Omega is a bounded smooth domain in R-N with N >= 3, (-Delta)(s) is the fractional Laplacian with s is an element of (0, 1), lambda > 0 is a parameter, alpha(x) : Omega -> R is a measurable function, F(x, u) : Omega x R -> R is a nonsmooth potential, and partial derivative F(x, u) is the generalized gradient of F(x, center dot) at u is an element of 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.