## 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.

### 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