  We study the existence and non-existence of positive solutions for the
$(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2} u + \beta
|u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in
$\Omega$. The main result of our research is the construction of a continuous
curve in $(\alpha,\beta)$ plane, which becomes a threshold between the
existence and non-existence of positive solutions. Furthermore, we provide the
example of domains $\Omega$ for which the corresponding first Dirichlet
eigenvalue of $-\Delta_p$ is not monotone w.r.t. $p > 1$.
