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The present paper deals with the spectrum of a fourth order nonlinear eigenvalue problem involving variable exponent conditions and a sign-changing potential. The main result of this paper establishes the existence of two positive constants $\lambda_0$ and $\lambda_1$ with $\lambda_0\leq \lambda_1$ such that every $\lambda\in [\lambda_1, +\infty)$ is an eigenvalue, while $\lambda\in (-\infty,\lambda_0)$ cannot be an eigenvalue of the above problem.