Uniqueness of the ground state solutions of \(\Delta{} u+f(u)=0\) in \(R^ n,n\geq{} 3\). (English) Zbl 0753.35034

The uniqueness question of positive radial solutions is studied for the following semilinear elliptic equations: \[ \begin{cases} u''(r)+{n-1 \over r}u'(r)+f(u)=0 & (r\geq 0) \\ u(r)>0 \text{ for }r\geq 0,\quad\text{and} & u(r)\to 0\text{ as }r\to +\infty \\ u'(0)=0. \end{cases}\tag{*} \] For the case \(f(u)=u^ p-u\), where \(1<p<{n+2 \over n-2}\) (\(n\geq 3\)) the uniqueness question of (*) was studied by Ch. V. Coffman [Arch. Ration. Mech. Anal. 46, 81-95 (1972; Zbl 0249.35029)], K. McLeod and J. Serrin [ibid. 99, 115-145 (1987; Zbl 0667.35023)], and finally solved by Man Kam Kwong [ibid 105, No. 3, 243-266 (1989; Zbl 0676.35032)].
In the paper the uniqueness of (*) is established for a more general class of \(f(u)\) including \(f(u)=u^ p+u^ q-2u\), where \(1<q<p\leq{n\over n-2}\). Part of the work overlaps with recent works by Kwong and Zhang, and McLeod.


35J65 Nonlinear boundary value problems for linear elliptic equations
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[2] DOI: 10.1512/iumj.1981.30.30012 · Zbl 0522.35036 · doi:10.1512/iumj.1981.30.30012
[3] DOI: 10.1007/BF00250684 · Zbl 0249.35029 · doi:10.1007/BF00250684
[4] Man Kam Kwong, Arch. Rational Mech. Analysis 105 pp 243– (1989)
[5] Kwong M. K., preprint
[6] McLeod K., preprint
[7] DOI: 10.1007/BF00275874 · Zbl 0667.35023 · doi:10.1007/BF00275874
[8] DOI: 10.1002/cpa.3160380105 · Zbl 0581.35021 · doi:10.1002/cpa.3160380105
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