Non-existence of nonnegative separate variable solutions to a porous medium equation with spatially dependent nonlinear source
Résumé
The non-existence of nonnegative compactly supported classical solutions to $−\Delta V (x) − |x|^\sigma V (x) + V^{1/m}(x)/(m − 1) = 0$, $x\in R^N$, with $m > 1$, $\sigma > 0$, and $N \ge 1$, is proven for $\sigma$ sufficiently large. More precisely, in dimension $N \ge 4$, the optimal lower bound on $\sigma$ for non-existence is identified, namely $\sigma \ge \sigma_c := 2(m − 1)(N − 1)/(3m + 1)$, while, in dimensions $N \in {1, 2, 3}$, the lower bound derived on $\sigma$ improves previous ones already established in the literature. A by-product of this result is the nonexistence of nonnegative compactly supported separate variable solutions to a porous equation medium equation with spatially dependent superlinear source.
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