Radially symmetric solutions of a nonlinear elliptic equation
Summary: We investigate the existence and asymptotic behavior of positive solutions of \[ w'' + ((N - 1)/r)w' - |w|^{p-1}w = 0\quad r > 0. \] We focus on the parameter regime \(N > 2\) and \(1 < p < N/(N - 2)\) where the equation has the closed form, positive singular solution \(w_1 = (4 - 2(N - 2)(p - 1)/(p - 1)^2)^{1/(p - 1)}r^{-2/(p - 1)}\), \(r > 0\). Our advance is to develop a technique to efficiently classify the behavior of solutions which are positive on a maximal positive interval \((r_{\min}, r_{\max})\). Our aim is to transform the nonautonomous \(w\) equation into an autonomous ODE. This reduces the problem to analyzing the behavior of solutions in the phase plane of the autonomous equation. We then show how specific solutions of the autonomous equation give rise to the existence of several new families of singular solutions. Specifically, we prove the existence of a family of singular solutions which exist on the entire interval \((0, \infty)\), and which satisfy \(0 < w(r) < w_1(r)\) for all \(r > 0\). An important open problem for the nonautonomous equation is presented. Its solution would lead to the existence of a new family of ``super singular'' solutions which lie entirely above \(w_1(r)\).
scientific article; zbMATH DE number 5930213