  We study in detail the existence, nonexistence and behavior of global
minimizers, ground states and corresponding energy levels of the
$(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u + \beta
|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under zero
Dirichlet boundary condition, where $p > q > 1$ and $\alpha, \beta \in
\mathbb{R}$. A curve on the $(\alpha,\beta)$-plane which allocates a set of the
existence of ground states and the multiplicity of positive solutions is
constructed. Additionally, we show that eigenfunctions of the $p$- and
$q$-Laplacians under zero Dirichlet boundary condition are linearly
independent.
