  We study the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u
= \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset
\mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical
curves on the $(\alpha,\beta)$-plane corresponding to the threshold of
existence of special classes of positive solutions. In particular, in certain
neighbourhoods of the point $(\alpha,\beta) = \left(\|\nabla
\varphi_p\|_p^p/\|\varphi_p\|_p^p, \|\nabla
\varphi_p\|_q^q/\|\varphi_p\|_q^q\right)$, where $\varphi_p$ is the first
eigenfunction of the $p$-Laplacian, we show the existence of two and, which is
rather unexpected, three distinct positive solutions, depending on a relation
between the exponents $p$ and $q$.
