Existence results for a class of degenerate elliptic equations

Paolo Caldiroli, SISSA via Beirut, 2-4 I 34013 Trieste

                   paolocal@sissa.it

We investigate the existence of positive solutions to Dirichlet problems of the form

−div(a(x)∇u) = f(x, u) in Ω,

where Ω is a domain in ℝ^{n}, a is a non negative function which vanishes somewhere and may be unbounded at infinity, and f is a smooth function such that sf(x, s) ∼ |s|^{p} as |s| → ∞, being p > 1. According to the behavior of a at its zeroes and at infinity, we can distinguish a subcritical case and a critical case, and in both situations we obtain some existence results. In particular, we study the critical case for the model equation

−div(|x|α∇u) = |u|^{2α^{∗}−2}u,

where 2α^{∗} = \frac{2n}{n−2+α}, and we prove that a positive ground state solution exists for the corresponding Dirichlet problem on a domain Ω whenever Ω is a cone, including the case Ω = ℝ^{n}. Moreover, concerning the case of an arbitrary domain Ω, we show how the geometry of Ω near the origin and at infinity affects the existence or non existence of ground state solutions.