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

Abstract

Let $p$ and $q$ be locally Hölder functions in $\RR^N$, $p>0$ and $q\ge 0$. We study the Emden-Fowler equation $-\Delta u+q(x)|\nabla u{|}^a=p(x)u^{-\gamma }$ 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.