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We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smoothbounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$then, for every positive, finite Borel measure $\mu$ on $\partial \Omega$,there exists a solution of (E) such that $u=\mu$ on $\partial \Omega$.Furthermore, if $N(p+q-1)\geq p+1$ then an isolated point singularity on$\partial \Omega$ is removable. In particular there is no solution withboundary data $\delta_y$ (=Dirac measure at a point $y\in \partial \Omega$).Finally we obtain a classification of positive solutions with an isolatedboundary singularity.

Positive solutions of quasilinear elliptic equations with subquadratic growth in the gradient