The authors study the Dirichlet problem for the equation \[ - \sum^{n}_{i=1}(\partial /\partial x_ i)a_ i(x,u,\nabla u)+f(x,u,\nabla u)=0 \] in the Sobolev space \(W^{1,p}(\Omega)\), \(\Omega\) a bounded domain in \(R^ n\) and \(p>1\). The coefficients are assumed to satisfy the usual coercivity and monotonicity conditions and, moreover, such natural growth conditions that the problem becomes meaningful in the class \(W^{1,p}(\Omega)\cap L^{\infty}(\Omega)\). In particular, the lower order term f(x,\(\eta\),\(\xi)\) is allowed to grow as fast as \(\sum a_ i(x,\eta,\xi)\xi_ i\) for \(| \xi | \to +\infty\), namely like \(| \xi |^ p.\)
Under mild continuity assumptions on the coefficients the authors prove the existence of a solution \(u\in W_ 0^{1,p}(\Omega)\cap L^{\infty}(\Omega)\) provided there exist a Lipschitzian subsolution \(\phi\) and a Lipschitzian supersolution \(\psi\) with \(\phi\leq \psi\) and \(\phi =\psi =0\) on \(\partial \Omega\). As the authors state, no such result was known before under equally mild regularity assumptions on the coefficients and at the same time allowing the natural growth of f.