Nonexistence Results for Elliptic Problems with Supercritical Nonlinearity in Nontrivial Domains

Let Ω be a bounded domain in Rn with n≥3 and assume p>2n/(n−2). We study problems of the type Δu+|u|p−2u=0 in Ω, u=0 on ∂Ω. We prove that for suitable p≥2n(n −2) there exists some bounded domain Ω with nontrivial topology (in the sense of Bahri and Coron) such that the above problem has only the solution u≡0.