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

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

in with has no positive -solution. Here we require .

Indeed, to see this, we first denote

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