| Summary: | In this article, we consider the nonlinear eigenvalue problem:
$$\left\{
\begin{array}{ll}
\Delta(|\Delta u|^{p_1(x)-2}\Delta u)+\Delta(|\Delta u|^{p_2(x)-2}\Delta u)=\lambda|u|^{q(x)-2}u,\ {\rm in}\ \Omega, & \\
u=\Delta u=0,\ {\rm on}\ \partial \Omega,\\
\end{array}
\right.$$
where $\Omega$ is a bounded domain of $\mathbb{R}^N$ with smooth boundary, $\lambda$ is a positive real number, the continuous functions $p_1,p_2$, and $q$ satisfy $1<p_2(x)<q(x)<p_1(x)<\frac{N}{2}$ and $\max\limits_{y\in\overline{\Omega}}q(y)<\frac{Np_2(x)}{N-2p_2(x)}$ for any $x\in \overline{\Omega}$. The maim result of this paper establishes the existence of two positive constants $\lambda_0$ and $\lambda_1$ with $\lambda_0\leq\lambda_1$ such that any $\lambda\in[\lambda_1,+\infty)$ is an eigenvalue, while and $\lambda\in(0,\lambda_0)$ is not an eigenvalue of the above problem.
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