Electron. J. Diff. Equ., Vol. 2014 (2014), No. 86, pp. 1-12.

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

Nemat Nyamoradi, Nguyen Thanh Chung

In this article, we study the existence and multiplicity of solutions to the nonlocal Kirchhoff fractional equation
 \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 \text{in }   \Omega,\cr
 u = 0 \quad \text{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.

Submitted December 15, 2013. Published April 2, 2014.
Math Subject Classifications: 34B27, 35J60, 35B05.
Key Words: Kirchhoff nonlocal operators; fractional differential equations; genus properties; critical point theory.

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Nemat Nyamoradi
Department of Mathematics, Faculty of Sciences
Razi University, 67149 Kermanshah, Iran
email: nyamoradi@razi.ac.ir, neamat80@yahoo.com
Nguyen Thanh Chung
Department of Mathematics, Quang Binh University
312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam
email: ntchung82@yahoo.com

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