Let $\Omega$ be $\mathbb R^n$ or a complete non-compact manifold, we consider $$\Delta u+f\cdot u+u^2\leq0,$$ where $\Delta$ denotes $-\sum^n_{i=1}\partial^2_{x_i}$ and $f$ is a $C^2$ function such that $|f|\leq C_0$.

Suppose $u\geq0,~u\in L^1(\Omega)$.

**Q** Is there any paper or research to consider the estimate of the $\|u\|_{L^1}$ of such an inequality.