(untitled) (Journal of Partial Differential Equations, 2020)
We consider the problem of whether the equation Δu = p(x) f (u) on R^N, N ≥ 3, has a positive solution for which lim_|x|→∞ u(x)=∞ where f is locally Lipschitz continuous, positive, and nondecreasing on (0,∞) and satisfies ∫^∞_1[F(t)]^-1/2dt=∞ where F(t)=∫^t_0f(s)ds. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies ∫^∞_0r_|x|=rp(x)dr=∞. Conversely, we show that a necessary condition for the solution to exist is that p satisfies ∫^∞_0r^1+ε_|x|=rp(x)dr=∞ for all ε > 0.