The author proves the existence of a weak solution for \[ - \sum^{\infty}_{i=1}\partial_ i[a_ i(x,u(x),\text{grad} u(x))]/\partial x_ i+a_ 0(x)u(x)+f\quad (x,u(x),\text{grad} u(x))=0,\quad x\in \Omega \]
\[ u=0\quad on\quad \partial \Omega, \] if \(\Omega\) is an unbounded domain with smooth boundary. The “divergence term” differs from a linear operator with constant coefficients by a differential operator supported in a compact subset of \(\Omega\), and f has quadratic growth with respect to grad u. Moreover, a stability result is given.