Radially symmetric solutions of \(\Delta w + |w|^{p-1}w = 0\) (Q1953669)

Summary: We investigate solutions of \[ w'' + ((N - 1)/r)w' + |w|^{p-1}w = 0, r > 0 \] 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 \((r_{\min}, r_{\max})\), 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, \(N/(N - 2) < p < (N + 2)/(N - 2)\), 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.