Structure of positive radial solutions to the Haraux-Weissler equation. (English) Zbl 0934.34028

Dohmen, Claus; Hirose, Munemitsu

Nonlinear Anal., Theory Methods Appl. 33, No. 1, 51-69 (1998).

The authors consider the system \[ u_{rr}+{n-1\over r} u_r+{r \over 2}u_r+ \lambda u+|u|^{p-1}u=0,\;r>0,\;u(0)=\alpha>0, \tag{1} \] with \(n\geq 3\), \(p>1\), \(\lambda>0\). The main result is the following: Suppose \(n\geq 3\), \(1<p<(n+2)/(n-2)\). If \(0<\lambda\leq(n-2)/2\), then there exists a unique positive number \(\alpha_\lambda\) such that \(u(r;\alpha_\lambda)\) is a rapidly decaying solution. Morover, \(u(r,\alpha)\) is a crossing solution for every \(\alpha \in(\alpha_\lambda,\infty)\) and a slowly decaying solution for every \(\alpha\in (0,\alpha_\lambda)\).

Reviewer: S.Mazanik (Minsk)

MSC:

34C11	Growth and boundedness of solutions to ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems, general theory

Keywords:

selfsimilar solutions; uniqueness; rapidly decaying solution

References:

[1]	Atkinson, F.V.; Peletier, L.A., Sur LES solutions radiales de l’equation \(Δu + (x · ∇u)2 + λu2 + |u|\^{}\{p−1\}u = 0\), C. R. acad. sci. Paris, serie I, 302, 99-101, (1986)
[2]	Escobedo, M.; Kavian, O., Variational problems related to self-similar solutions of the heat equation, Nonlinear analysis, 11, 1103-1133, (1987) · Zbl 0639.35038
[3]	Haraux, A.; Weissler, F.B., Nonuniqueness for a semilinear initial value problem, Indiana univ. M. J., 31, 167-189, (1982) · Zbl 0465.35049
[4]	Hirose, M., Structure of positive radial solutions to a semilinear elliptic PDE with a gradient-term, Funkc. ekvac., 39, 323-345, (1996) · Zbl 0863.34002
[5]	Peletier, L.A.; Terman, D.; Weissler, F.B., On the equation \(Δu + (x · ∇u)2 + ƒ(u) = 0\), Arch. rat. mech. anal., 94, 83-99, (1986) · Zbl 0615.35034
[6]	Weissler, F.B., Asymptotic analysis of an ODE and nonuniqueness for a semilinear PDE, Arch. rat. mech. anal., 91, 231-245, (1986) · Zbl 0614.35043
[7]	Weissler, F.B., Rapidly decaying solutions of an ODE with application to semilinear elliptic on parabolic pdes, Arch. rat. mech. anal., 91, 247-266, (1986) · Zbl 0604.34034
[8]	Yanagida, E., Uniqueness of rapidly decaying solutions to the haraux-weissler equation, J. diff. eqs., 127, 561-570, (1996) · Zbl 0856.34058
[9]	Yanagida, E. and Yotsutani, S., A unified approach to the structure of radial solutions to semilinear elliptic problems. (In Preparation.) · Zbl 1001.35041
[10]	Yotsutani, S., Pohozaev identity and its applications, Kyoto university Sûrikaisekikenkyûsho Kôkyûroku, 834, 80-90, (1993)