Radially Symmetric Solutions of Δw+wp−1w=0

Abstract

We investigate solutions of $w^{\prime \prime }+\left(\left(N-1\right)/r\right)w^{\prime }+{\left|w\right|}^{p-1}w=0,\hspace{0.17em}r>0$ and focus on the regime $N>2$ and $p>N/(N-2)$. Our advance is to develop a technique to efficiently classify the behavior of solutions on $(r_{\text{min}},r_{\text{max}})$, their maximal positive interval of existence. Our approach is to transform the nonautonomous $w$ equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the $w$ equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, $w_1$, such that $w_1(r)\to \infty$ as $r\to 0^{+}$ and $w_1(r)\to 0$ as $r\to \infty$. Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies$w(r_i)=w_1(r_i)$ for infinitely many values $r_i>0$. At the critical value $p=(N+2)/(N-2)$ there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime $p>(N+2)/(N-2)$ we prove the existence of a family of “super singular” sign changing singular solutions.