Abstract. We study the existence and non-existence of ground states for the Schrödinger equations −∆u − λ. . , xm) ∈ R mN , and −∆u − λu/|y| 2 = |u| 2 * −2 u, x = (y, z) ∈ R N . In both cases we assume λ = 0 and λ < λ, where λ is the Hardy constant corresponding to the problem.

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We prove two Hardy-Sobolev type inequalities in D 1,2 (R N), resp. in H 1 0 (Ω), where Ω is a bounded domain in R N , N ≥ 3. The framework involves the singular potential |x| −a , with a ∈ (0, 1). Our paper extends previous results established by Bianchi and Egnell ([2]), resp. by Brezis and Lieb ([3]), corresponding to the case a = 0.

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