The authors study the nonlinear boundary value problem \[ \text{div }A(-\nabla u)=f(x,u,\nabla u)\quad\text{in }\Omega, \qquad u=0\quad\text{on }\partial\Omega,\tag{\(*\)} \] where \(\text{div }A(-\nabla u)\) is a nonlinear operator of \(p\)-Laplacian type. They prove existence of a nontrivial solution of this problem in the following cases: \[ (1)\quad f(x,u,\nabla u)=h-u|\nabla u|^{p-2}, \qquad (2)\quad f(x,u,\nabla u)=g(x,u). \] Here, \(h\in L^\infty\) and \(g\) is superlinear and subcritical in \(u\). In the first case, an approximation procedure is used, where \(u|\nabla u|^{p-2}\) is replaced by \(u|\nabla u|^{p-2}/(1+ \varepsilon|u|\;|\nabla u|^{p-2})\). The corresponding solution, \(u_\varepsilon\), is shown to converge to a solution of \((*)\) as \(\varepsilon\to0\), For case (2), the mountain pass lemma is used to construct a pair of nontrivial solutions \(u_-\leq 0\leq u_+\).