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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:
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