Uniqueness of positive radial solutions of \(\Delta u+f(u)=0\) in \(\mathbb{R}^ n\). II. (English) Zbl 0804.35034

McLeod, Kevin

Trans. Am. Math. Soc. 339, No. 2, 495-505 (1993).

Summary: [For part I see Arch. Ration. Mech. Anal. 99, 115-145 (1987; Zbl 0667.35023).]
We prove a uniqueness result for the positive radial 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 \(f(x) = - u + u^ p\), \(1<p<(n + 2)/(n - 2)\). The result is proved by reducing to an initial-boundary problem for the ODE \(u'' + (n - 1)/r + f(u) = 0\) and using a shooting method.

MSC:

35J60	Nonlinear elliptic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

uniqueness; positive radial solution

Citations:

Zbl 0667.35023