Infinitely many solutions for hemivariational inequalities involving the fractional Laplacian

Abstract

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

Keywords

• Fractional Laplacian
• Hemivariational inequality
• Nonsmooth analysis
• Infinitely many solutions