Let Ω be a bounded domain in R n , we prove the singular Moser-Trudinger embedding: sup u ≤1 Ω e α|u| n n−1 |x| β < ∞ if and only if α αn + β n ≤ 1, where α > 0, β ∈ [0, n), u ∈ W 1,n 0 (Ω) and u = Ω |∇u| n 1 n. We will also study the corresponding critical exponent problem.2000 Mathematics Subject Classification: 35B33, 35J20, 35J60.

We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

In this paper we study the existence, non-existence and simplicity of the¯rst eigenvalue of the perturbed Hardy{Sobolev operator ¡¢ ¡ 1 4 (n ¡ 2) 2 (q=jxj 2 ) under various assumptions on the perturbation q. We study the asymptotic behaviour of the¯rst eigenfunction near the origin when the perturbation q is q = s, 0 < s < 1. We will also establish the best constant in a Hardy{Sobolev inequality proved by Adimurthi e t a l .

We prove that a sharp Moser-Trudinger inequality holds true on a conformal disc if and only if the metric is bounded from above by the Poincaré metric. We also derive necessary and sufficient conditions for the validity of a sharp Moser Trudinger inequality on a simply connected domain in R 2

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