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