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

We investigate the existence and multiplicity of abstract weak solutions of the equation \(-\Delta _p u -\Delta _q u=\alpha |u|^{p-2}u+\beta |u|^{q-2}u\) $-{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_{q}u={\alpha |u|}^{p-2}u+\beta {|u|}^{q-2}u$ in a bounded domain under zero Dirichlet boundary conditions, assuming \(1<q<p\) $1<q<p$ and \(\alpha ,\beta \in {\mathbb {R}}\) $\alpha ,\beta \in \mathbb{R}$ . We determine three generally different ranges of parameters \(\alpha \) $\alpha$ and \(\beta \) $\beta$ 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 smaller and larger constraint sets than in the standard minimax definition.