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In this note, we deal with the existence of infinitely many solutions for a problem driven by nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions \begin{equation*} \begin{cases} -\mathcal{L}_{K}u=\lambda f(x,u), & {\rm in}\ \Omega, \\ u=0, &{\rm in}\;\mathbb{R}^{n}\backslash\Omega, \\ \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain of $\mathbb{R}^{n}$ and the nonlinear term $f$ satisfies superlinear at infinity but does not satisfy the the Ambrosetti–Rabinowitz type condition. The aim is to determine the precise positive interval of $\lambda$ for which the problem admits at least two nontrivial solutions by using abstract critical point results for an energy functional satisfying the Cerami condition.