FAST DECAYING POSITIVE SOLUTIONS FOR div(|\nabla u|^{p-2} \nabla u ) + f(|x|)u^{-\gamma}=0} IN R^n, n > 2; A CONJECTURE OF FURUSHO ET AL
Abstract
For $ p \in (1, n)$, $\gamma>0 $ and a non negative $ f \in C( [0, \infty))$, it is shown in \cite 2 that the problem (P) below has a fast decaying radial solution if: \medskip
1) the function $ t^\nu f(t) $ is in $ L^1(\Bbb R_+) $ for $ \nu:= n-1 + \gamma(n-p)/(p-1) $, \medskip
\noindentand \medskip
2) $ \gamma \in (0, p-1)$.\medskip
The authors also conjecture that $ \gamma>0 $ needs not be less than $ p-1 $ for that result to hold. In this note, via a method of comparison, we prove that conjecture and also we indicate that the condition $ f(0)>0 $ is not necessary.
1) the function $ t^\nu f(t) $ is in $ L^1(\Bbb R_+) $ for $ \nu:= n-1 + \gamma(n-p)/(p-1) $, \medskip
\noindentand \medskip
2) $ \gamma \in (0, p-1)$.\medskip
The authors also conjecture that $ \gamma>0 $ needs not be less than $ p-1 $ for that result to hold. In this note, via a method of comparison, we prove that conjecture and also we indicate that the condition $ f(0)>0 $ is not necessary.
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