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Remarks on the uniqueness of radial solutions. (English) Zbl 0697.35046
Uniqueness of radial solutions for the problem \(\Delta u+f(u)=0\) in \(D=B_ R(0)\), with \(au-b(\partial u/\partial n)=0\) on \(\partial D\) is considered. Here a and b are constants and n denotes the outward normal unit vector. As usual, the problem is reduced to a second order ordinary differential equation for the radial solution u(r), \(r=| x|\), \(0<r<R\), and the associated system is studied with phase plane arguments. The function f is \(C^ 1\) and, besides some more technical assumptions, it is supposed that f has a hyperbolic zero c \((f(c)=0\), \(f'(c)<0)\). Uniqueness follows from the properties of the time-map arising in the problem; more precisely, if the time-map \(T_ k\) is such that \(T'_ k>0\) in a left-neighborhood of c, where k is a positive integer, then radial solutions are locally unique in each k-nodal class. This fact has important implications concerning symmetry-breaking. Proofs are only sketched, they involve rather sophisticated phase plane arguments. Full proofs and more details can be found in the following paper of the authors [J. Differ. Equations 77, No.2, 287-303 (1989; Zbl 0673.34008)].
Reviewer: J.Hernandez
MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:
[1] P. CLEMENT and G. SWEERS, Existence and multiplicity results for a semilinear eigenvalue problem (preprint).
[2] J. SMOLLER and A. WASSERMAN, Existence, uniqueness, and nondegeneracy of positive solutions of semilineare elliptic equations, Comm. Math. Phys., 95 (1984), 129-159. Zbl0582.35046 MR760329 · Zbl 0582.35046 · doi:10.1007/BF01468138
[3] J. SMOLLER and A. WASSERMAN, Symmetry, degeneracy, and universahty in semilinear elliptic equations, I. Infinitesimal symmetry breaking (to appear in J. Funct. Anal.). Zbl0702.35016 MR1042215 · Zbl 0702.35016 · doi:10.1016/0022-1236(90)90099-7
[4] J. SMOLLER and A. WASSERMAN, On the monotonicity of the time map J. Diff. Eqns. Zbl0673.34008 · Zbl 0673.34008 · doi:10.1016/0022-0396(89)90145-9
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