Abstract. For Ω ⊂ R n , n ≥ 2, a bounded domain, and for 1 < p < n, we improve the Hardy-Sobolev inequality by adding a term with a singular weight of the type () 2 . We show that this weight function is optimal in the sense that the inequality fails for any other weight function more singular than this one. Moreover, we show that a series of finite terms can be added to improve the Hardy-Sobolev inequality, which answers a question of BrezisVazquez. Finally, we use this result to analyze the behaviour of the first eigenvalue of the operator Lµu := −(div(|∇u| p−2 ∇u) +

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In this paper we consider the semilinear elliptic problem in a bounded domain « R n ,where · > 0, 0 6 ¬ 6 2, 2 ¤ ¬ := 2(n ¡ ¬ )=(n ¡ 2), f : « ! R + is measurable, f > 0 a.e, having a lower-order singularity than jxj ¡ 2 at the origin, and g : R ! R is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions = p for which the embedding W 1;p 0 (« ) ,! L p (« ; f ) is compact. When p = 2, ¬ = 2, f 2 = 2 and 0 6 · < ( 1 2 (n ¡ 2)) 2 , we prove that the linear problem has H 1 0 -discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of = 2 , the¯rst eigenvalue goes to a positive number as · approaches ( 1 2 (n ¡ 2)) 2 . Furthermore, when g is superlinear, we show that for the same subclass of = 2 , the functional corresponding to the di® erential equation satis¯es the Palais{Smale condition if ¬ = 2 and a Brezis{Nirenberg type of phenomenon occurs for the case 0 6 ¬ < 2.

We prove that any distribution q satisfying the grad-div system ∇q = div f for some tensor-the local Hardy space; q is in h r and q is locally represented by the sum of singular integrals of f i j with Calderón-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure p (modulo constant) associated with incompressible elastic energy-minimizing deformation u satisfying |∇u| 2 , |cof ∇u| 2 ∈ h 1 . We also derive the system of Euler-Lagrange equations for volume preserving local minimizers u that are in the space K 1,3 loc [defined in (1.2)]-partially resolving a long standing problem. In two dimensions we prove partial C 1,α regularity of weak solutions provided their gradient is in L 3 and p is Hölder continuous.