We consider the eigenvalue problem $$ \begin{array}{l}\displaystyle-{\rm div} (a(|\nabla u |)\nabla u) = \lambda g(x, u) \;\mbox{in } \Omega u = 0\;\mbox{on } \partial\Omega ,\end{array}$$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.