Existence of solutions to nonlocal Kirchhoff equations of elliptic type via genus theory

Nyamoradi, Nemat ; Chung, Nguyen Thanh

Electronic Journal of Differential Equations, Tome 2014 (2014).

Résumé

Summary: In this article, we study the existence and multiplicity of solutions to the nonlocal Kirchhoff fractional equation $$\displaylines{ \Big(a + b\int_{\mathbb{R}^{2N}} |u (x) - u (y)|^2 K (x - y)\,dx\,dy\Big) (- \Delta)^s u - \lambda u = f (x, u (x)) \quad {in } \Omega,\cr u = 0 \quad {in } \mathbb{R}^N \setminus \Omega, }$$ where $a, b > 0$ are constants, $(- \Delta)^s$ is the fractional Laplace operator, $s \in (0, 1)$ is a fixed real number, $\lambda$ is a real parameter and $\Omega$ is an open bounded subset of $\mathbb{R}^N, N > 2 s$, with Lipschitz boundary, $f: \Omega \times \mathbb{R} \to \mathbb{R}$ is a continuous function. The proofs rely essentially on the genus properties in critical point theory.

Détail

Classification : 34B27, 35J60, 35B05
Keywords: Kirchhoff nonlocal operators, fractional differential equations, genus properties, critical point theory