Structure Results for Semilinear Elliptic Equations with Hardy Potentials

Abstract

Abstract We prove structure results for the radial solutions of the semilinear problem Δ ⁢ u + λ ⁢ ( | x | ) | x | 2 ⁢ u + f ⁢ ( u ⁢ ( x ) , | x | ) = 0 , \Delta u+\frac{\lambda(|x|)}{|x|^{2}}u+f(u(x),|x|)=0, where λ is a function and f is superlinear in the u-variable. As particular cases, we are able to deal with Matukuma potentials and with nonlinearities f having different polynomial behaviors at zero and at infinity. We give the complete picture for the subcritical, critical and supercritical cases. The technique relies on the Fowler transformation, allowing to deal with a dynamical system in ℝ 3 {{\mathbb{R}}^{3}} , for which elementary invariant manifold theory allows to draw the conclusions involving regular/singular and fast/slow-decay solutions.