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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.

MSC:

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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References:

[1] Berestycki H., Arch. Rational Mech. Analysis 82 pp 313– (1983)
[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
[9] Ni W. M., II 8, in: Rend. Circolo Mat. Palermo (Centenary Supplement) pp 171– (1985)
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