Entire solutions of singular elliptic equations (Q1822682)

This paper is concerned with the existence and asymptotic properties of solutions to the singular elliptic equation \[ -\Delta u=f(x)u^{- \lambda},\quad \lambda >0. \] It is assumed that f is positive and locally \(\alpha\)-Hölder continuous in \(R^ n\). The main results give sufficient conditions for the existence of positive, entire solutions having asymptotic behavior \[ (i)\quad u(x)\sim | x|^{2- n},\quad n\geq 3;\quad (ii)\quad u(x)\sim \log (x),\quad n=2. \] The method is based on the Schauder fixed point theorem and classical integral operator equations in \(R^ n\). It is of particular interest because it gives better information about rate of decay as \(| x| \to \infty\), and it generalizes to higher order equations where methods based on the maximum principle do not apply.