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: ( −4pu = λ|u|r−2u+ μ |u| q−2 |x|s u in Ω, u|∂Ω = 0, where λ and μ are two positive parameters and Ω is a smooth bounded domain in Rn containing 0 in its interior. The variational approach requires that 1 < p < n, p ≤ q ≤ p∗(s) ≡ n−s n−pp and p ≤ r ≤ 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−pp. 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 ≤ p).