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

Abstract

The non-existence of nonnegative compactly supported classical solutions to $$-\Delta V(x)-|x{|}^{\sigma }V(x)+\frac{V^{1/m}(x)}{m-1}=0,x\in {\mathbb{R}}^N,$$ 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 $$\sigma \ge {\sigma }_c:=\frac{2(m-1)(N-1)}{3m+1},$$ 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 non-existence of nonnegative compactly supported separate variable solutions to a porous equation medium equation with spatially dependent superlinear source.