Bubble Concentration on Spheres for Supercritical Elliptic Problems

Pacella, Filomena; Pistoia, Angela

doi:10.1007/978-3-319-04214-5_20

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…They used Proposition 3.2 to establish the existence of positive and sign changing solutions to problem (℘ 2 * N,k −ε ) concentrating along the (m − 1)-dimensional spheresS 1 := {(x, 0) ∈ R m × {0} : |x| = a}, S 2 := {(0, y) ∈ {0} × R m : |y| = a}as ε → 0. Their results can be found in[20].…”

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confidence: 94%

Symmetries, Hopf fibrations and supercritical elliptic problems

Clapp¹,

Pistoia²

2016

Contemporary Mathematics

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We consider the semilinear elliptic boundary value problem −∆u = |u| p−2 u in Ω, u = 0 on ∂Ω, in a bounded smooth domain Ω of R N for supercritical exponents p > 2N N −2 . Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both.The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents.

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confidence: 94%

Symmetries, Hopf fibrations and supercritical elliptic problems

Clapp¹,

Pistoia²

2016

Contemporary Mathematics

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“…A fruitful approach to produce solutions to the supercritical problem (1.1) is to reduce it to some critical or subcritical problem in a domain of lower dimension, either by considering rotational symmetries, or by means of maps which preserve the laplacian, or by a combination of both. This approach has been recently taken in [1,4,11,12,15,21] to produce solutions of (1.1) in different types of domains. We shall also follow this approach to obtain a new type of solutions in domains with thin spherical perforations.…”

Section: Introductionmentioning

confidence: 99%

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

Clapp

Faya

Pistoia

2015

JAMA

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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.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Clapp

Ghimenti

Micheletti

2014

Journal of Mathematical Analysis and Applications

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Abstract. Given a smooth Riemannian manifold (M, g) we investigate the existence of positive solutions to the equationwhich concentrate at some submanifold of M as ε → 0, for supercritical nonlinearities. We obtain a posive answer for some manifolds, which include warped products.Using one of the projections of the warped product or some harmonic morphism, we reduce this problem to a problem of the formwith the same exponent p, on a Riemannian manifold (M, h) of smaller dimension, so that p turns out to be subcritical for this last problem. Then, applying Lyapunov-Schmidt reduction, we establish existence of a solution to the last problem which concentrates at a point as ε → 0.

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Bubble Concentration on Spheres for Supercritical Elliptic Problems

Cited by 4 publications

References 24 publications

Symmetries, Hopf fibrations and supercritical elliptic problems

Symmetries, Hopf fibrations and supercritical elliptic problems

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

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

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