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Semilinear elliptic equations with near critical growth rates

Published online by Cambridge University Press:  14 November 2011

C. Budd
C. Budd
Affiliation:
Mathematical Institute, 24/29 St. Giles, Oxford OX1 3LB, U.K.
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Synopsis

We discuss the symmetric solutions of the semilinear elliptic equation Δu + λ(u+ u|u|p−1) = 0, u|∂B = 0 (*), where B is the unit ball in ℝ3. The value of p is taken close to 5, the critical Sobolev exponent for ℝ3. An asymptotic description of the solutions of (*) with large norm is obtained. This predicts a fold bifurcation if p > 5 and the structure of this bifurcation is studied in the limit p – 5→ 0. We find good agreement between the asymptotic description and some numerical calculations. These results are illuminated by recasting the problem (*) in the form of a dynamical system by means of a suitable change of variables. When |p – 5|≪1 and ∥u ≫1, the transformed solutions of (*) are also solutions of a perturbed Hamiltonian system and we study the behaviour of these solutions by using Melnikov methods.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 107 , Issue 3-4 , 1987 , pp. 249 - 270
Copyright
Copyright © Royal Society of Edinburgh 1987

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