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