Positive solutions to a supercritical elliptic problem that concentrate along a thin spherical hole

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 126; no. 1; pp. 341 - 357
• Main Authors: Clapp, Mónica, Faya, Jorge, Pistoia, Angela
• Format: Journal Article
• Language: English
• Published: Jerusalem The Hebrew University Magnes Press 01.04.2015; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Analysis; Dynamical Systems and Ergodic Theory; Functional Analysis; Mathematics; Mathematics and Statistics; Partial Differential Equations; Texts; Invariant Solution; Closed Subgroup; Smooth Domain; Elliptic Problem; Harmonic Morphism
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/s11854-015-0020-6

We consider the supercritical problem , where Θ is a bounded smooth domain in ℝ N , N ≥ 3, p > 2*:= 2 N /( N − 2), and Θ ∈ is obtained by deleting the ∈ -neighborhood of some sphere which is embedded in Θ. We show that in some particular situations, for small enough ∈ > 0, this problem has a positive solution {itv}{in{it\te}} and that this solution concentrates and blows up along the sphere as ∈ → 0. Our approach is to reduce this problem by means of a Hopf map to a critical problem of the form , in a punctured domain of lower dimension. We show that if Ω is a bounded smooth domain in ℝ n , n ≥ 3, is positive, and , then for small enough ∈ > 0, this problem has a positive solution u ε which concentrates and blows up at Ξ 0 as ∈ → 0.