We investigate the existence of ground state solutions to the Dirichlet problem -div(|x|α∇u) = |u|2*α-2u in Ω, u = 0 on ∂Ω, where α ε (0,2), 2*α = 2n/n-2+α and Ω is a domain in R^n. In particular we prove that a non negative ground state solution exists when the domain Ω is a cone, including the case Ω = R^n. 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.
On the existence of extremal functions for a weighted Sobolev embedding with critical exponent
Musina, Roberta
1999-01-01
Abstract
We investigate the existence of ground state solutions to the Dirichlet problem -div(|x|α∇u) = |u|2*α-2u in Ω, u = 0 on ∂Ω, where α ε (0,2), 2*α = 2n/n-2+α and Ω is a domain in R^n. In particular we prove that a non negative ground state solution exists when the domain Ω is a cone, including the case Ω = R^n. 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.File in questo prodotto:
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