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).

(more…)

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