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

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The non-existence of nonnegative finite energy solutions to − Δ V ( x ) − | x | σ V ( x ) + V 1 / m ( x ) m − 1 = 0 , x ∈ R N , with m > 1 , σ > 0 , and N ≥ 1 , is proven for σ sufficiently large. More precisely, in dimension N ≥ 4 , the optimal lower bound on σ for non-existence is identified, namely σ ≥ σ c : = 2 ( m − 1 ) ( N − 1 ) 3 m + 1 , while, in dimensions N ∈ { 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 medium equation with spatially dependent superlinear source.