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

We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: [GRAPHICS] where lambda and mu are two positive parameters and Omega is a smooth bounded domain in R-n containing 0 in its interior. The variational approach requires that 1 < p< n, p less than or equal to q less than or equal to p*(s) = n-s/n-p p and p less than or equal to r less than or equal to p* = p*(0) = np/n-p, which we assume throughout. However, the situations differ widely with q and r, and the interesting cases occur either at the critical Sobolev exponent (r = p*) or in the Hardy-critical setting (s = p = q) or in the more general Hardy-Sobolev setting when q = n-s/n-p p. 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 p = 2, especially those corresponding to singularities (i.e., when 0 < s less than or equal to p).