scholarly journals An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent

2011 ◽  
Vol 4 (4) ◽  
pp. 907-922 ◽  
Author(s):  
Futoshi Takahashi ◽  
Keyword(s):  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Critical Sobolev Exponent ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic ◽  
Blowing Up ◽  
Sobolev Exponent ◽  
Blowing Up Solutions

Semilinear elliptic equations with near critical growth rates

1987 ◽  
Vol 107 (3-4) ◽  
pp. 249-270 ◽  
Author(s):  
C. Budd
Keyword(s):  
Elliptic Equations ◽  
Critical Sobolev Exponent ◽  
Semilinear Elliptic Equation ◽  
Asymptotic Description ◽  
Change Of Variables ◽  
Semilinear Elliptic ◽  
Fold Bifurcation ◽  
Sobolev Exponent ◽  
Suitable Change ◽  
Good Agreement

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.

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