Let $\Omega \subset R^m $ be a bounded domain and ${\bf L}$ a second order uniformly strongly elliptic partial differential operator. Let ${\bf B}$ be a linear boundary operator. Suppose $f(x,u)$ and $g(x,u)$ are functions on $\Omega \times R^1 $ which are nonincreasing with respect to u for sufficiently large values of $| u |$. Conditions are found under which the problem $Lu = f(x,u(x))$, $x \in \Omega $, with $Bu = g(x,u(x))$, $x \in \partial \Omega $, has a generalized solution in the Sobolev space $H^1 (\Omega )$. This is followed by a brief discussion of stability of positive solutions.