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

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Let p and q be locally Hölder functions in ⁿ, p>0 and q≥ 0. We study the Emden-Fowler equation -Δu+ q(x)|∇ u|ᵃ=p(x)u⁻γ in ⁿ, where a and γ 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.