  We consider a $(p,2)$-equation, that is, a nonlinear nonhomogeneous elliptic
equation driven by the sum of a $p$-Laplacian and a Laplacian with $p>2$. The
reaction term is $(p-1)$-linear but exhibits asymmetric behaviour at
$\pm\infty$ and at $0^{\pm}$. Using variational tools, together with truncation
and comparison techniques and Morse theory, we prove two multiplicity theorems,
one of them providing sign information for all the solutions (positive,
negative, nodal).
