Liouville–type theorems for semilinear elliptic equations involving the Sobolev exponent

1998 ◽  
Vol 228 (4) ◽  
pp. 723-744 ◽  
Author(s):  
Chang-Shou Lin
Keyword(s):  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Liouville Type ◽  
Semilinear Elliptic ◽  
Liouville Type Theorems ◽  
Sobolev Exponent

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.

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.

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