A smooth boundary domain \(\Omega\) in \(R^N\) and the real numbers \(p,q, \mu\), verifying \(2 \leq q < p < \infty\), \( \mu \geq 0\), are considered. A qualitative analysis of the equation \[ -\mathrm{div}(|\nabla u|^{p-2}\nabla u)-\mu\cdot \mathrm{div}(|\nabla u|^{q-2}\nabla u)=f(x,y) \quad \text{in} \quad \Omega \] with homogeneous Dirichet boundary conditions on \(\partial \Omega\) is performed. It is proved that the above problem possesses at least three solutions in \(C_0^1(\overline{\Omega})\). A nodal solution exists, between two others with constant sign, when the reaction term \(f(x,y)\) has a subcritical growth and is verifying some limit behaviors related with the eigenvalues of the \(p\) and \(q\) Laplacian. The hypothesis on \(f\) gives a corresponding coercive energy functional. Palais-Smale condition, sequentially lower-semicontinuity, some compactness properties and the Mountain-Pass theorem are used to obtain the results.