alexa On the existence of extremal functions for a weighted Sobolev embedding with critical exponent


Journal of Astrophysics & Aerospace Technology

Author(s): Paolo Caldiroli, Roberta Musina

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We investigate the existence of ground state solutions to the Dirichlet problem −÷(|x|α∇u)=|u|2∗α−2u−÷(|x|α∇u)=|u|2α∗−2u in ΩΩ , u = 0 on ∂Ω∂Ω , where α∈(0,2)α∈(0,2) , 2∗α=2nn−2+α2α∗=2nn−2+α and ΩΩ is a domain in RnRn . In particular we prove that a non negative ground state solution exists when the domain ΩΩ is a cone, including the case Ω=RnΩ=Rn . Moroever, we study the case of arbitrary domains, showing how the geometry of the domain near the origin and at infinity affects the existence or non existence of ground state solutions.

This article was published in Calculus of Variations and Partial Differential Equations and referenced in Journal of Astrophysics & Aerospace Technology

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