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

Abstract

We consider the supercritical problemwhere Θ is a bounded smooth domain in R N , N ≥ 3, p > 2 * := 2N N−2 , and Θǫ is obtained by deleting the ǫ-neighborhood of some sphere which is embedded in Θ. In some particular situations we show that, for ǫ > 0 small enough, this problem has a positive solution vǫ and that these solutions concentrate and blow up along the sphere as ǫ → 0.Our approach is to reduce this problem to a critical problem of the formin a punctured domain Ωǫ := {x ∈ Ω : |x − ξ 0 | > ǫ} of lower dimension, by means of some Hopf map. We show that, if Ω is a bounded smooth domain in R n , n ≥ 3, ξ 0 ∈ Ω, Q ∈ C 2 (Ω) is positive and ∇Q(ξ 0 ) = 0 then, for ǫ > 0 small enough, this problem has a positive solution uǫ, and that these solutions concentrate and blow up at ξ 0 as ǫ → 0.Key words: Nonlinear elliptic problem; supercritical problem; nonautonomous critical problem; positive solutions; domains with a spherical perforation, blow-up along a sphere.MSC2010: 35J60, 35J20.

Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents

“…This is a typical phenomenon for concentration on positive dimensional sets. We point out that the case k 2 under some symmetric assumptions on the domain was studied by Ackermann-Clapp-Pistoia in [1], see also [9] for some related issues.…”

Symmetries, Hopf fibrations and supercritical elliptic problems

“…Note that G may be the trivial group, so this result is true in a non-symmetric setting and, combined with Proposition 3.1, yields solutions to supercritical problems concentrating around a spherical hole, see [9].…”

“…[16] and the references therein. This method was used by Faya and the authors in [9] to prove the following result. Theorem 3.6.…”

“…But we may also combine it with Proposition 3.2 as follows:Let N = 2m, m ≥ 2, Ω be an [O(m) × O(m)]-invariant bounded smooth domain in R N ≡ R m × R m such that 0 / ∈ Ω, and ξ ∈ Ω ∩ (R m × {0}) . For ε > 0 small enough set Ω ε := {x ∈ Ω : dist(x, S ξ ) > ε}, where S ξ := {(x, 0) ∈ R m × {0} : |x| = |ξ|}.The following result, obtained by combining Theorem 3.6 with Proposition 3.2, was established in[9]. For each ε small enough the supercritical problem…”

Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold

“…This approach was introduced by Ruf and Srikanth in [26], where the reduction is given by a Hopf map. Reductions may also be performed by means of other maps which preserve the laplacian, or by considering rotational symmetries, or by a combination of both, as has been recently done in [1,11,12,19,20,24,25,30] for different problems.…”