# Ngô Quốc Anh

## December 11, 2016

### Why the equation Δu+1/u^α=0 in R^n has no positive solution?

Filed under: Uncategorized — Ngô Quốc Anh @ 23:50

Following the idea due to Brezis in his note that we had already mentioned once, it seems that we can show that the following equation

$\displaystyle \Delta u + \frac 1{u^\alpha}=0$

in $\mathbb R^n$ with $n\geqslant 2$ has no positive $C^2$-solution. Here we require $\alpha>0$.

Indeed, to see this, we first denote

$f(t)=-t^{-\alpha}$

with $t>0$. Clearly the function $f$ is monotone increasing in its domain. Using this, we rewrite our equation as follows

$\Delta u =f(u).$