  We investigate the existence of nodal (sign-changing) solutions to the
Dirichlet problem for two-parametric family of partially homogeneous
$(p,q)$-Laplace equations $-\Delta_p u -\Delta_q u=\alpha |u|^{p-2}u+\beta
|u|^{q-2}u$ where $p \neq q$. By virtue of the Nehari manifolds, linking
theorem, and descending flow, we explicitly characterize subsets of
$(\alpha,\beta)$-plane which correspond to the existence of nodal solutions. In
each subset the obtained solutions have prescribed signs of energy and, in some
cases, exactly two nodal domains. The nonexistence of nodal solutions is also
studied. Additionally, we explore several relations between eigenvalues and
eigenfunctions of the $p$- and $q$-Laplacians in one dimension.
