Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\). The author uses variational methods to deduce sufficient conditions for the existence and multiplicity of weak solutions of the quasilinear Dirichlet problem: \[ -\text{div} \biggl(a \bigl(|\nabla u|^p \bigr)|\nabla u|^{p-2} \nabla u\biggr) =f(x,u) \quad \text{in }\Omega,\quad u=0 \quad\text{on } \partial\Omega, \] where \(p>1\). Such problems arise in the study of non-Newtonian fluids and nonlinear elasticity. Certain structure conditions are imposed on the functions \(a\) and \(f\), e.g., \(f\) is a Carathéodory function with subcritical growth among other restrictions. The case of the \(p\)-Laplacian is included since \(a(s)\equiv 1\) is allowed. The functional used in this work is \[ J(u)= {1\over p} \int_\Omega A\bigl(|\nabla u|^p \bigr)dx- \int_\Omega F(x,u)dx, \] where \(A\) and \(F\) are primitives of \(a\) and \(f\), respectively, and the detailed structure conditions imply a compactness result of Palais-Smale type. Several specific examples illustrate the results.