Add to ORKGWe 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.Comment: 29 pages, 3 figure
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