Existence of three solutions for Kirchhoff nonlocal operators of elliptic type
Existence of three solutions for Kirchhoff nonlocal operators of elliptic type
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发表时间：
2013-10
影响因子：
0.4
通讯作者：
N. Nyamoradi
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文献类型：
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作者：
N. Nyamoradi
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].
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].