  We investigate the existence and the multiplicity of solutions of the problem
$$ \begin{cases} -\Delta_p u-\Delta_q u = g(x, u)\quad & \mbox{in } \Omega,\\
\displaystyle{u=0} & \mbox{on } \partial\Omega, \end{cases} $$ where $\Omega$
is a smooth, bounded domain of $\mathbb R^N$, $1<p<q<\infty$, and the
nonlinearity $g$ behaves as $u^{q-1}$ at infinity. We use variational methods
and find multiple solutions as minimax critical points of the associated energy
functional. Under suitable assumptions on the nonlinearity, we cover also the
resonant case.
