A Supercritical Elliptic Problem in a Cylindrical Shell

Clapp, Mónica; Szulkin, Andrzej

doi:10.1007/978-3-319-04214-5_13

Cited by 4 publications

(3 citation statements)

References 18 publications

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“…Concerning the supercritical problem (℘ λ ) with k ≥ 1, Passaseo [16,17] showed that a nontrivial solution does not exist if λ = 0 and Θ is a ball. This statement was extended in [5] to more general domains Θ, and to some unbounded domains in [6]. On the other hand, existence of multiple solutions has been established in [4,14,23].…”

Section: Introductionmentioning

confidence: 88%

Ground states of critical and supercritical problems of Brezis–Nirenberg type

Clapp

Pistoia

Szulkin

2016

Annali di Matematica

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Abstract. We study the existence of symmetric ground states to the supercritical problemin a domain of the formwhere Θ is a bounded smooth domain such thatis the (k + 1)-st critical exponent. We show that symmetric ground states exist for λ in some interval to the left of each symmetric eigenvalue, and that no symmetric ground states exist in some interval (−∞, λ * ) with λ * > 0 if k ≥ 2.Related to this question is the existence of ground states to the anisotropic critical problemwhere a, b, c are positive continuous functions on Θ. We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of λ, and obtain a bifurcation result for ground states.

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Section: Introductionmentioning

confidence: 88%

Ground states of critical and supercritical problems of Brezis–Nirenberg type

Clapp

Pistoia

Szulkin

2016

Annali di Matematica

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“…We now return to the Dirichlet problems. There have been many supercritical works that deal with domains that have certain symmetry, for instance, see [15,16,17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

Cowan¹,

Moameni²

2021

Preprint

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In this paper we are interested in positive classical solutions ofwhere Ω is a bounded annular domain (not necessarily an annulus) in R N (N ≥ 3) and a(x) is a nonnegative smooth function. We show the existence of a classical positive solution for all p > 2 when the problem enjoys certain mild symmetry conditions. As a consequence of our results, we shall show that (1) has a nonradial solution when a(x) = 1 and Ω is an annulus with certain assumptions on the radii. Our approach is based on a new variational principle that allows one to deal with supercritical problems variationally by limiting the corresponding functional on a proper convex subset instead of the whole space.

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“…This follows immediately from Proposition 4.1 using standard variational methods because problem (4.2) is subcritical and the domain Θ is bounded. Existence and nonexistence results in some unbounded domains are also available [11].…”

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confidence: 99%

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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A Supercritical Elliptic Problem in a Cylindrical Shell

Abstract: We consider the problem −∆u = |u| p−2 u in Ω, u = 0 on ∂Ω,We show that 2 * N,m is the true critical exponent for this problem, and that there exist nontrivial solutions if 2 < p < 2 * N,m but there are no such solutions if p ≥ 2 * N,m .

Cited by 4 publications

References 18 publications

Ground states of critical and supercritical problems of Brezis–Nirenberg type

Ground states of critical and supercritical problems of Brezis–Nirenberg type

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

Symmetries, Hopf fibrations and supercritical elliptic problems

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