| By a sub-supersolution method and a perturbed argument, we show the existence of entire solutions for the semilinear elliptic problem $- \Delta u +a(x)|\nabla u|^q=\lambda b(x)g(u)$, $u>0$, $x\in \mathbb R^N$, $\lim_{|x|\rightarrow \infty} u(x)=0$, where $q\in (1,2]$, $\lambda>0$, $a$ and $b$ are locally H\"{o}lder continuous, $a\geq 0$, $b>0$, $\forall x\in \mathbb R^N$, and $g\in C^1((0,\infty), (0,\infty))$ which may be both possibly singular at zero and strongly unbounded at infinity. |
