Existence of three solutions for Kirchhoff nonlocal operators of elliptic type
Abstract
In this paper we prove the existence of at least three solutionsto 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].
Keywords
Nonlocal fractional equation, Nonsmooth critical point, Variational methods, Locally Lipschitz, Three solutions.