Structure results for semilinear elliptic equations with Hardy potentials (Q680357)

This article contributes to the existing literature with the classification of positive solutions for singular ODE's \[ \displaystyle u'' + \frac{n-1}{r}u'+\frac{\lambda(r)}{r^2}u +f(u,r)=0, \;\;\;r\in(0,\infty), \] for a wide range of potentials \(\lambda(r)/r^2\) and nonlinearities \(f(u,r)\). It is assumed that \(n> 2\). Such ODE's are obtained by symmetry reduction from the corresponding elliptic PDE's in \(\mathbb{R}^n\) with radial coefficients. The authors build on the known cases \(\lambda\) constant and \(\displaystyle f(u,r) = cr^\delta u|u|^{q-2}\) with \(q >2\) and \(\delta > -2\). General results are stated and proved for classes of nonlinearities and potentials, and more specific examples are given as corollaries. Most of the proofs are done via phase portrait analysis after converting the equation to a first order system.