Non-existence of nonnegative separate variable solutions to a porous medium equation with spatially dependent nonlinear source

The non-existence of nonnegative compactly supported classical solutions to $- ΔV(x) - |x|^σV(x) + \frac{V^{1/m}(x)}{m-1} = 0, \qquad x\in\mathbb{R}^N,$ with $m>1$, $σ>0$, and $N\ge 1$, is proven for $σ$ sufficiently large. More precisely, in dimension $N\geq4$, the optimal lower bound on $σ$ for non-existence is identified, namely $σ\geqσ_c := \frac{2(m-1)(N-1)}{3m+1},$ while, in dimensions $N\in\{1,2,3\}$, the lower bound derived on $σ$ improves previous ones already established in the literature. A by-product of this result is the non-existence of nonnegative compactly supported separate variable solutions to a porous equation medium equation with spatially dependent superlinear source.

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