Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents

Authors:
N. Ghoussoub and C. Yuan

Journal:
Trans. Amer. Math. Soc. **352** (2000), 5703-5743

MSC (2000):
Primary 35J20, 35J70, 47J30, 58E30

DOI:
https://doi.org/10.1090/S0002-9947-00-02560-5

Published electronically:
July 6, 2000

MathSciNet review:
1695021

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation:

where and are two positive parameters and is a smooth bounded domain in containing in its interior. The variational approach requires that , and , which we assume throughout. However, the situations differ widely with and , and the interesting cases occur either at the critical Sobolev exponent () or in the Hardy-critical setting () or in the more general Hardy-Sobolev setting when . In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen *dual sets*. Many of the results are new even in the case , especially those corresponding to singularities (i.e., when .

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Additional Information

**N. Ghoussoub**

Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

**C. Yuan**

Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada

DOI:
https://doi.org/10.1090/S0002-9947-00-02560-5

Received by editor(s):
August 11, 1998

Published electronically:
July 6, 2000

Article copyright:
© Copyright 2000
American Mathematical Society