In this paper, we are concerned with the following general nonlocal problem \begin{equation*} -\mathcal{L}_K u=\lambda_1u+f(x,u)\ \ \text{in}\ \Omega,
u=0\ \ \text{in}\ \mathbb{R}^N\backslash\Omega, \end{equation*} where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded and with continuous boundary.

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