  We investigate the existence and multiplicity of abstract weak solutions of
the equation $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in
a bounded domain under zero Dirichlet boundary conditions, assuming $1<q<p$ and
$\alpha,\beta \in \mathbb{R}$. We determine three generally different ranges of
parameters $\alpha$ and $\beta$ for which the problem possesses a given number
of distinct pairs of solutions with a prescribed sign of energy. As auxiliary
results, which are also of independent interest, we provide alternative
characterizations of variational eigenvalues of the $q$-Laplacian using
narrower and larger constraint sets than in the standard minimax definition.
