Infinitely many solutions for hemivariational inequalities involving the fractional Laplacian

Abstract

AbstractIn the paper, we consider the following hemivariational inequality problem involving the fractional Laplacian: $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda u\in \alpha (x) \partial F(x,u) & x \in \varOmega , \\ u=0 & x\in \mathbb{R} ^{N} \backslash \varOmega , \end{cases} $${(−Δ)su+λu∈α(x)∂F(x,u)x∈Ω,u=0x∈RN∖Ω, where Ω is a bounded smooth domain in $\mathbb{R} ^{N}$RN with $N\geq 3$N≥3, $(-\Delta )^{s}$(−Δ)s is the fractional Laplacian with $s\in (0,1)$s∈(0,1), $\lambda >0$λ>0 is a parameter, $\alpha (x): \varOmega \rightarrow \mathbb{R} $α(x):Ω→R is a measurable function, $F(x, u):\varOmega \times \mathbb{R} \rightarrow \mathbb{R} $F(x,u):Ω×R→R is a nonsmooth potential, and $\partial F(x,u)$∂F(x,u) is the generalized gradient of $F(x, \cdot )$F(x,⋅) at $u\in \mathbb{R} $u∈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.