A Paneitz–Branson type equation with Neumann boundary conditions

Bonheure, Denis; Ali, Hussein Cheikh; Nascimento, Robson

doi:10.1515/acv-2019-0023

Cited by 3 publications

(7 citation statements)

References 61 publications

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“…We recall that the constant S appearing in [4] (see eq. ( 2.1) therein) corresponds to our S 2 1, N +4 N −4…”

Section: Let Us Definementioning

confidence: 99%

“…We now prove that Σ is attained. We borrow some ideas from [4,Lemma 3.2]. We preliminary notice that…”

Section: Let Us Definementioning

confidence: 99%

“…Proof. Following the proof of (1.8) of Lemma 2.1 in [4] (see Step two therein), we choose finitely many points x i ∈ Ω and a positive number R > 0, with corresponding sets…”

Section: A Cherrier's-type Inequalitiesmentioning

confidence: 99%

“…In particular, this shows the existence of solutions to (1.4) also in the dimensions N = 5, 6, however the proof of this result actually works for all N ≥ 5, therefore it also gives a different proof of existence in case N ≥ 7. Although, up to our knowledge, apparently there are no results regarding (1.4) in the literature, we recall that in [4] a related problem with a Paneitz-Branson type operator, Lw := ∆ 2 u − ∆u + αu with α > 0, and the same boundary conditions was studied.…”

Section: Introductionmentioning

confidence: 99%

“…Section 5 is devoted to the study of the biharmonic equation (1.4), and to the proof of Theorem 1. 4. In Section 6 we analyze the symmetry-breaking phenomenon in the annulus, and existence of solutions with radial symmetry.…”

Section: Introductionmentioning

confidence: 99%

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Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

Pistoia¹,

Schiera²,

Tavares³

2022

Preprint

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We study the following Lane-Emden systemwith Ω a bounded regular domain of R N , N ≥ 4, and exponents p, q belonging to the so-called critical hyperbola 1/(p + 1) + 1/(q + 1) = (N − 2)/N . We show that, under suitable conditions on p, q, leastenergy (sign-changing) solutions exist. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If N ≥ 5, p = 1, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, if Ω is an annulus, we prove that least energy solutions are not radially symmetric, although there exist radial ones for any p, q > 0.

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“…We recall that the constant S appearing in [4] (see eq. ( 2.1) therein) corresponds to our S 2 1, N +4 N −4…”

Section: Let Us Definementioning

confidence: 99%

“…We now prove that Σ is attained. We borrow some ideas from [4,Lemma 3.2]. We preliminary notice that…”

Section: Let Us Definementioning

confidence: 99%

“…Proof. Following the proof of (1.8) of Lemma 2.1 in [4] (see Step two therein), we choose finitely many points x i ∈ Ω and a positive number R > 0, with corresponding sets…”

Section: A Cherrier's-type Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

Pistoia¹,

Schiera²,

Tavares³

2022

Preprint

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Existence of Solutions on the Critical Hyperbola for a Pure Lane–Emden System with Neumann Boundary Conditions

Pistoia

Schiera

Tavares

2023

International Mathematics Research Notices

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We study the following Lane–Emden system: $$\begin{align*} & -\Delta u=|v|^{q-1}v \quad \ \textrm{in}\ \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \ \textrm{in}\ \Omega, \qquad u_{\nu}=v_{\nu}=0 \quad \ \textrm{on}\ \partial \Omega, \end{align*}$$with $\Omega $ a bounded regular domain of ${\mathbb{R}}^{N}$, $N \ge 4$, and exponents $p, q$ belonging to the so-called critical hyperbola $1/(p+1)+1/(q+1)=(N-2)/N$. We show that, under suitable conditions on $p, q$, least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation, which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier-type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If $N \ge 5$, $p=1$, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case $\Omega $ is a ball or an annulus.

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Topics in elliptic problems: from semilinear equations to shape optimization

Tavares

2024

Communications in Mathematics

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In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text.

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A Paneitz–Branson type equation with Neumann boundary conditions

Cited by 3 publications

References 61 publications

Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions

Existence of Solutions on the Critical Hyperbola for a Pure Lane–Emden System with Neumann Boundary Conditions

Topics in elliptic problems: from semilinear equations to shape optimization

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