Existence of three solutions for Kirchhoff nonlocal operators of elliptic type

Abstract

In this paper we prove the existence of at least three solutions
to the following Kirchhoff nonlocal fractional equation:
\begin{equation*}
\begin{cases}
M \left (\int_{\mathbb{R}^n\times \mathbb{R}^n} |u (x) - u (y)|^2 K (x - y) d x d y - \int_\Omega |u (x)|^2 d x
\right) ((- \Delta)^s u - \lambda u) \\
\hspace{2cm}
\in \theta (\partial j (x, u (x)) + \mu \partial k (x, u (x))),
& \textrm{in}\;\;   \Omega,\\
u = 0, & \textrm{in}\;\;   \mathbb{R}^n \setminus \Omega,
\end{cases}
\end{equation*}
where $(- \Delta)^s$ is the fractional Laplace operator, $s \in
(0, 1)$ is a fix, $\lambda, \theta, \mu$ are real parameters and
$\Omega$ an open bounded subset of $\mathbb{R}^n$, $n > 2 s$, with
Lipschitz boundary. The approach is fully based on a recent three
critical points theorem of Teng [K. Teng, Two nontrivial solutions
for hemivariational inequalities driven by nonlocal elliptic
operators, Nonlinear Anal. (RWA) 14 (2013) 867-874].