Abstract multiplicity results for $(p,q)$-Laplace equations with two parameters

We investigate the existence and multiplicity of abstract weak solutions of the equation $-{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_{q}u=\alpha \mathrm{\mid }u{\mathrm{\mid }}^{p-2}u+\beta \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u$ 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.