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In this article, we study the existence and multiplicity of solutions to the nonlocal Kirchhoff fractional equation $$\displaylines{ \Big(a + b\int_{\mathbb{R}^{2N}} |u (x) - u (y)|^2 K (x - y)\,dx\,dy\Big) (- \Delta)^s u - \lambda u = f (x, u (x)) \quad \text{in } \Omega,\cr u = 0 \quad \text{in } \mathbb{R}^N \setminus \Omega, }$$ 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.