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.

  • $\begingroup$ 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 $\endgroup$ – Math604 Mar 5 at 20:18
  • $\begingroup$ whoops...didn't see $\Delta$ is really $ -\Delta$.... $\endgroup$ – Math604 Mar 5 at 20:29
  • $\begingroup$ do you know anymore about $f$ ? positive, negative ??? $\endgroup$ – Math604 Mar 5 at 20:31
  • $\begingroup$ 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 $N$. $\endgroup$ – Math604 Mar 6 at 15:01
  • $\begingroup$ @Math604 I have seen any paper in your case considering the $L^1$-estimate, could you give one reference. $\endgroup$ – DLIN Mar 7 at 6:03

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