The author proves the existence and uniqueness of a weak solution in certain weighted Sobolev spaces for the Dirichlet problem \[ \begin{aligned} L u(X) &=f_0(x)-\sum_{j=1}^n D_j f_j(x) \text{ in }\Omega, \\ u(x) &=0 \text{ on }\partial \Omega, \end{aligned} \] with the partial differential operator \(L\) defined as \[ L u(x)=\Delta \Bigl( v(x) |\Delta u|^{p-2} \Delta u\Bigr) - \sum_{j=1}^n D_j [\omega (x) \mathcal{A}_j(x, u(x), \nabla u(x))]. \] The set \(\Omega\) is a bounded open set, \(\omega\) and \(v\) are two weight functions of the Muckenhoupt class, \(1<p<\infty\), and the functions \(\mathcal{A}_j\) satisfy certain regularity and growth assumptions. The author proves by a topological argument (uniqueness of solution of the equation \(Ax=T\) with strictly monotone, coercive, and hemicontinuous operator \(A\) on the real, separable, reflexive Banach space) that the problem has a unique solution and provides an estimate for its norm.