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

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We study the asymptotic behavior of radial solutions for a singularly perturbed semilinear elliptic Dirichlet problem on an annulus. We show that Morse index informations on such solutions provide a complete description of the blow-up behavior. As a by-product, we exhibit some sufficient conditions to guarantee that radial ground state solutions blow-up and concentrate at the inner/outer boundary of the annulus

We review some recent results concernig existence/ non existence/ uniqueness of extremals for Sobolev inequalities in Hyperbolic spaces. We also discuss exponential integrability in the hyperbolic plane and related topics.Mathematics Subject Classification (2010). 35J20, 35J60.

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