Entire positive solutions of the singular Emden-Fowler equation with nonlinear gradient term

Let $p$ and $q$ be locally H"{o}lder functions in ${\text{RR}}^N$, $p>0$ and $q\geq 0$\mathrm{.}WestudytheEmden-Fowlerequation$-\Delta u+ q(x)|\nabla u|^a=p(x)u^{-\gamma}$in$\RR^N$,where$a$and$\gamma$ are positive numbers. Our main result establishes that the above equation has a unique positive solutions decaying to zero at infinity. Our proof is elementary and it combines the maximum principle for elliptic equations with a theorem of Crandall, Rabinowitz and Tartar.