Structure of positive radial solutions of semilinear elliptic equations (Q678051)

This article primarily concerns uniqueness and asymptotic behavior of positive radial solutions to the semilinear problems (1) \(-\Delta u=f(u)\) in \(\mathbb{R}^n\), \(u(\infty)=0\); and (2) \(-\Delta u=f(u)\) in \(B\), \(u|_{\partial B}=0\), where \(B\) denotes a ball in \(\mathbb{R}^n\) centred at the origin, and \(f(t)= \min\{t^p,t^q\}\) for \(1<p< (n+2)/(n-2)<q\), \(q\geq 3\). The main theorems are: (a) Problem (2) has a unique positive radial solution in \(B\); (b) Problem (1) has an infinitude of positive radial solutions; (c) exactly one of the solutions in (b) has the asymptotic decay \(u(x)\sim c|x|^{2-n}\) as \(|x|\to\infty\), for some positive constant \(c\); and (d) all other positive solutions to (1) have the slower asymptotic decay \(u(x)\sim c^*|x|^{2/(1-q)}\) as \(|x|\to\infty\), where the positive constant \(c^*\) depends only on \(n\) and \(q\). The proofs employ a shooting method to an IVP for the polar form of \(-\Delta u=f(u)\). Earlier related results were obtained by \textit{B. Gidas}, \textit{Wei-ming Ni} and \textit{L. Nirenberg} [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)], \textit{L. A. Peletier} and \textit{J. Serrin} [J. Differ. Equations 61, 380-397 (1986; Zbl 0577.35035)], \textit{K. McLeod} and \textit{J. Serrin} [Arch. Ration. Mech. Anal. 99, 115-145 (1987; Zbl 0667.35023)], \textit{E. Yanagida} [Jap. J. Indust. Appl. Math. 8, No. 1, 165-173 (1991; Zbl 0735.35064)], \textit{M. K. Kwong} and \textit{Y. Li} [Trans. Am. Math. Soc. 333, No. 1, 339-363 (1992; Zbl 0785.35038)], and others cited therein.