A Riccatian approach to the decay of solutions of certain semi-linear PDE’s

We use a simple and effective Hille-type technique to establish that the nn-th-dimensional elliptic equation Δu+f(x,u)+g(|x|)x⋅∇u=0Δu+f(x,u)+g(|x|)x⋅∇u=0, |x|>A>0|x|>A>0, with n≥3n≥3, has positive solutions which decay to 0 as |x|→+∞|x|→+∞.