  We study the boundary value problem $-{\rm div}((|\nabla u|^{p\_1(x)
-2}+|\nabla u|^{p\_2(x)-2})\nabla u)=f(x,u)$ in $\Omega$, $u=0$ on
$\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\RR^N$. We
focus on the cases when $f\_\pm
(x,u)=\pm(-\lambda|u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where
$m(x):=\max\{p\_1(x),p\_2(x)\} < q(x) < \frac{N\cdot m(x)}{N-m(x)}$ for any
$x\in\bar\Omega$. In the first case we show the existence of infinitely many
weak solutions for any $\lambda>0$. In the second case we prove that if
$\lambda$ is large enough then there exists a nontrivial weak solution. Our
approach relies on the variable exponent theory of generalized Lebesgue-Sobolev
spaces, combined with a $\ZZ\_2$-symmetric version for even functionals of the
Mountain Pass Lemma and some adequate variational methods.
