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We study the nonlinear eigenvalue problem $-{\rm div}(a(|\nabla u|)\nablau)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$is a bounded open set in $\RR^N$ with smooth boundary, $q$ is a continuousfunction, and $a$ is a nonhomogeneous potential. We establish sufficientconditions on $a$ and $q$ such that the above nonhomogeneous quasilinearproblem has continuous families of eigenvalues. The proofs rely on elementaryvariational arguments. The abstract results of this paper are illustrated bythe cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.

A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces