# Estimate of $L^1$-norm of semilinear elliptic inequality

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.

• assume $u>0$ and divide equation by $u$ and then use standard ieqnuality to see something like $\| \nabla \phi \|_{L^2}^2 \ge \int f \phi^2 dx + \int u \phi^2 dx$ for all $\phi$ compactly supported.... this gives something...maybe – Math604 Mar 5 at 20:18
• whoops...didn't see $\Delta$ is really $-\Delta$.... – Math604 Mar 5 at 20:29
• do you know anymore about $f$ ? positive, negative ??? – Math604 Mar 5 at 20:31
• If you take $f=0$ and take equality; then try googling Keller - Osserman condition. This equation fits into that framework and i think there are `large solutions' on $R^N$ depending on the $p$ and $N$. – Math604 Mar 6 at 15:01
• @Math604 I have seen any paper in your case considering the $L^1$-estimate, could you give one reference. – DLIN Mar 7 at 6:03