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On the monotonicity of the time-map. (English) Zbl 0673.34008
The authors deal with the equation $$\Delta u(x)+f(u(x))=0,$$ $$| x| <R$$ together with the boundary conditions $$\alpha u(x)-\beta du(x)/dn=0,$$ $$| x| =R$$, and investigate the monotonicity of the time map $$p\mapsto T(p)$$, $$p=u(0)$$ which plays an important role in the behaviour of radical solutions and in symmetry breaking.
Reviewer: E.Barvinek

##### MSC:
 34A30 Linear ordinary differential equations and systems, general 34K05 General theory of functional-differential equations 35B32 Bifurcations in context of PDEs
##### Keywords:
monotonicity of the time map; radical solutions
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##### References:
 [1] \scP. Clement and G. Sweers, Existence and multiplicity results for a semilinear eigenvalue problem, preprint. · Zbl 0662.35045 [2] Smoller, J.; Wasserman, A., Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations, Comm. math. phys., 95, 129-159, (1984) · Zbl 0582.35046 [3] Smoller, J.; Wasserman, A., Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch. rational mech. anal., 95, 217-225, (1986) · Zbl 0629.35040 [4] Smoller, J.; Wasserman, A., Symmetry breaking for solutions of semilinear elliptic equations with general boundary conditions, Comm. math. phys., 105, 415-441, (1986) · Zbl 0608.35004 [5] \scJ. Smoller and A. Wasserman, Symmetry, degeneracy, and universality in semilinear elliptic equations. I. Infinitesimal symmetry-breaking, J. Funct. Anal. to appear. · Zbl 0702.35016
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