Uniqueness of positive radial solutions of Δ𝑢+𝑓(𝑢)=0 in 𝑅ⁿ. II

McLeod, Kevin

doi:10.1090/S0002-9947-1993-1201323-X

Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\textbf {R}^ n$. II
HTML articles powered by AMS MathViewer

by Kevin McLeod PDF

Trans. Amer. Math. Soc. 339 (1993), 495-505 Request permission

Abstract:

We prove a uniqueness result for the positive solution of $\Delta u + f(u) = 0$ in ${\mathbb {R}^n}$ which goes to $0$ at $\infty$. The result applies to a wide class of nonlinear functions $f$, including the important model case $f(u) = - u + {u^p}$ , $1 < p < (n + 2)/(n - 2)$. The result is proved by reducing to an initial-boundary problem for the ${\text {ODE}}\;u'' + (n - 1)/r + f(u) = 0$ and using a shooting method.

References

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, DOI 10.1007/BF00250555
H. Berestycki, P.-L. Lions, and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\textbf {R}^{N}$, Indiana Univ. Math. J. 30 (1981), no. 1, 141–157. MR 600039, DOI 10.1512/iumj.1981.30.30012
Charles V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^{3}=0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal. 46 (1972), 81–95. MR 333489, DOI 10.1007/BF00250684
Xabier Garaizar, Existence of positive radial solutions for semilinear elliptic equations in the annulus, J. Differential Equations 70 (1987), no. 1, 69–92. MR 904816, DOI 10.1016/0022-0396(87)90169-0
Man Kam Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. MR 969899, DOI 10.1007/BF00251502
Kevin McLeod and James Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\textbf {R}^n$, Arch. Rational Mech. Anal. 99 (1987), no. 2, 115–145. MR 886933, DOI 10.1007/BF00275874
Kevin McLeod, W. C. Troy, and F. B. Weissler, Radial solutions of $\Delta u+f(u)=0$ with prescribed numbers of zeros, J. Differential Equations 83 (1990), no. 2, 368–378. MR 1033193, DOI 10.1016/0022-0396(90)90063-U
L. A. Peletier and James Serrin, Uniqueness of positive solutions of semilinear equations in $\textbf {R}^{n}$, Arch. Rational Mech. Anal. 81 (1983), no. 2, 181–197. MR 682268, DOI 10.1007/BF00250651

Eigenfunctions of the equation

5

Li Qun Zhang, Uniqueness of positive solutions to semilinear elliptic equations, Acta Math. Sci. (Chinese) 11 (1991), no. 2, 130–142 (Chinese). MR 1129746