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
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by Kevin McLeod

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

DOI: https://doi.org/10.1090/S0002-9947-1993-1201323-X

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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.

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Eigenfunctions of the equation

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