Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

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$\Omega$isasmoothboundeddomainin$\RR^N$. We focus on the cases when $f_\pm (x,u)=\pm(-\lambda|u|^{m(x)-2}u+|u|^{q(x)-2}u)$,where$m(x):=\max{p_1(x),p_2(x)} < q(x) < \frac{N\cdot m(x)}{N-m(x)}$forany$x\in\bar\Omega$. 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 $\text{ZZ}\mathrm{\_}2$-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.