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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
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
Full Text: DOI EuDML
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[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
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