scholarly journals PRESCRIBING A FOURTH ORDER CONFORMAL INVARIANT ON THE STANDARD SPHERE — PART I: A PERTURBATION RESULT

2002 ◽  
Vol 04 (03) ◽  
pp. 375-408 ◽  
Author(s):  
ZINDINE DJADLI ◽  
ANDREA MALCHIODI ◽  
MOHAMEDEN OULD AHMEDOU
Keyword(s):  
Elliptic Equation ◽  
Fourth Order ◽  
Critical Sobolev Exponent ◽  
Order Elliptic Equation ◽  
Conformal Invariant ◽  
Standard Sphere ◽  
Finite Dimensional ◽  
Fourth Order Elliptic Equation ◽  
Zero Degree ◽  
Sobolev Exponent

In this paper we study some fourth order elliptic equation involving the critical Sobolev exponent, related to the prescription of a fourth order conformal invariant on the standard sphere. We use a topological method to prove the existence of at least a solution when the function to be prescribed is close to a constant and a finite dimensional map associated to it has non-zero degree

scholarly journals Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation

2019 ◽  
Vol 22 (06) ◽  
pp. 1950057
Author(s):  
Zongming Guo ◽  
Fangshu Wan ◽  
Liping Wang
Keyword(s):  
Elliptic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Order Elliptic Equation ◽  
Liouville Type ◽  
Related Equation ◽  
Fourth Order Elliptic Equation ◽  
Liouville Type Results

New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth-order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.

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