  We study the boundary value problem $-{\rm div}((|\nabla
u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)=\lambda|u|^{q(x)-2}u$ in
$\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in
$\RR^N$ with smooth boundary, $\lambda$ is a positive real number, and the
continuous functions $p_1$, $p_2$, and $q$ satisfy $1<p_2(x)<q(x)<p_1(x)<N$ and
$\max_{y\in\bar\Omega}q(y)<\frac{N p_2(x)}{N-p_2(x)}$ for any $x\in\bar\Omega$.
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
any $\lambda\in[\lambda_1,\infty)$ is an eigenvalue, while any
$\lambda\in(0,\lambda_0)$ is not an eigenvalue of the above problem.
