Semilinear elliptic equations with near critical growth rates

Abstract

SynopsisWe 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.