## 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
- Trans. Amer. Math. Soc.
**339**(1993), 495-505 - DOI: https://doi.org/10.1090/S0002-9947-1993-1201323-X
- PDF | 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
—, Personal communication.
- 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
S. I. Pohozaev, - Li Qun Zhang,
*Uniqueness of positive solutions to semilinear elliptic equations*, Acta Math. Sci. (Chinese)**11**(1991), no. 2, 130–142 (Chinese). MR**1129746**

*Eigenfunctions of the equation*$\Delta u + \lambda f(u) = 0$, Soviet Math.

**5**(1965), 1408-1411.

## Bibliographic Information

- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**339**(1993), 495-505 - MSC: Primary 35J60; Secondary 34B15, 35B05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1201323-X
- MathSciNet review: 1201323