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

Abstract

We consider the supercritical problem

$$ - \Delta v = {\left| v \right|^{p - 2}}v{\text{ in }}{\Theta _\epsilon },{\text{ }}v = 0{\text{ on }}\partial {\Theta _\epsilon }$$

, where Θ is a bounded smooth domain in ℝ^N, N ≥ 3, p > 2*:= 2N/(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

$$ - \Delta u = Q(x){\left| u \right|^{4/n - 2}}u{\text{ in }}{\Omega _\epsilon },{\text{ }}u = 0{\text{ on }}\partial {\Omega _\epsilon }$$

, in a punctured domain ${\Omega _\epsilon }: = \{ x \in \Omega :\left| {x - {\xi _0}} \right| > \epsilon \}$ of lower dimension. We show that if Ω is a bounded smooth domain in ℝⁿ, n ≥ 3, ${\xi _0} \in \Omega ,Q \in {C^2}(\overline \Omega )$ is positive, and $\nabla Q({\xi _0}) \ne 0$, then for small enough ∈ > 0, this problem has a positive solution u _ε which concentrates and blows up at Ξ ₀ as ∈ → 0.