scholarly journals Bifurcation of Gradient Mappings Possessing the Palais-Smale Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Elliot Tonkes
Keyword(s):  
Elliptic Equations ◽  
Principal Eigenvalue ◽  
Critical Sobolev Exponent ◽  
Semilinear Elliptic Equations ◽  
Asymptotic Nature ◽  
Semilinear Elliptic ◽  
Palais Smale Condition ◽  
Sobolev Exponent

This paper considers bifurcation at the principal eigenvalue of a class of gradient operators which possess the Palais-Smale condition. The existence of the bifurcation branch and the asymptotic nature of the bifurcation is verified by using the compactness in the Palais Smale condition and the order of the nonlinearity in the operator. The main result is applied to estimate the asyptotic behaviour of solutions to a class of semilinear elliptic equations with a critical Sobolev exponent.

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.

The existence of a positive solution of semilinear elliptic equations with limiting Sobolev exponent

1991 ◽  
Vol 117 (1-2) ◽  
pp. 75-88 ◽  
Author(s):  
Shixiao Wang
Keyword(s):  
Positive Solution ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Thin Domain ◽  
Existence Of A Solution ◽  
Semilinear Elliptic ◽  
Sobolev Exponent ◽  
Image Position

SynopsisOur paper concerns the existence of a positive solution for the equation:A new condition, which guarantees the existence of a solution of the above equation, has been established. It has also given some sharp information in the cases where: (1) a(x) = λ = const. and Ω is a “thin” domain; (2) Ω is a ball and a(x) is a radially symmetrical function.

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