  We consider parametric equations driven by the sum of a $p$-Laplacian and a
Laplace operator (the so-called $(p,2)$-equations). We study the existence and
multiplicity of solutions when the parameter $\lambda>0$ is near the principal
eigenvalue $\hat{\lambda}_1(p)>0$ of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. We
prove multiplicity results with precise sign information when the near
resonance occurs from above and from below of $\hat{\lambda}_1(p)>0$.
