Structure theorems for positive radial solutions to \(\text{div}(| Du|^{m-2} Du)+K(| x|)u^ q=0\) in \(\mathbb{R}^ n\)
The aim of the paper is to characterize the structure of positive radial solutions of the quasilinear equation \[ \text{div} \bigl( | Du |^{m -2} Du \bigr) + K \bigl( | x | \bigr) u^ q = 0, \quad x \in \mathbb{R}^ n, \tag{E} \] where \(q>m-1\) and \(n>m>1\). For positive radial solutions, (E) is reduced to the initial value problem \[ \bigl( r^{n - 1} | u_ r |^{m-2} u_ r \bigr)_ r + r^{n-1} K(r) (u^ +)^ q = 0, \quad r>0, \quad u(0) = \alpha>0. \leqno (K_ \alpha) \] Under some growth conditions of a nonnegative function \(K(r)\), \((K_ \alpha)\) has a unique global solution \(u(r,\alpha)\) for every \(\alpha>0\). As for the behavior of \(u(r,\alpha)\) at \(r=\infty\), the structure of solutions of \((K_ \alpha)\) is classified into one of three types: (i) for every \(\alpha\), \(u(r, \alpha)\) has a finite zero, (ii) for every \(\alpha\), \(u(r,\alpha)\) remains positive and \(\lim_{r \to \infty}r^{(n - m)/(m - 1)} u(r,\alpha) = \infty\), (iii) there exists a positive number \(\alpha^*\) such that for \(\alpha \in (\alpha^*, \infty)\) and \(\alpha \in (0, \alpha^*)\), \(u(r,\alpha)\) behaves as in (i) and (ii) respectively, and \(u(r,\alpha^*)\) remains positive with a finite positive limit of \(r^{(n - m)/(m - 1)} u(r,\alpha^*)\) at \(r = \infty\). The semilinear version \((m=2)\) of the result has been established in [\textit{N. Kawano}, \textit{W.-M. Ni} and \textit{S. Yotsutani}, J. Math. Soc. Japan 42, No. 3, 541-564 (1990; Zbl 0714.35028)].
scientific article; zbMATH DE number 562719