  We consider the Dirichlet problem for the nonhomogeneous equation $-\Delta_p
u -\Delta_q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u + f(x)$ in a bounded
domain, where $p \neq q$, and $\alpha, \beta \in \mathbb{R}$ are parameters. We
explore assumptions on $\alpha$ and $\beta$ that guarantee the resolvability of
the considered problem. Moreover, we introduce several curves on the
$(\alpha,\beta)$-plane allocating sets of parameters for which the problem has
or does not have positive or sign-changing solutions, provided $f$ is of a
constant sign.
