Abstract

The non-existence of nonnegative finite energy solutions to−ΔV(x)−|x|σV(x)+V1/m(x)m−1=0,x∈RN, 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, namelyσ≥σc:=2(m−1)(N−1)3m+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.

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