Multi-layer radial solutions for a supercritical Neumann problem

Bonheure, Denis; Grossi, Massimo; Noris, Benedetta; Terracini, Susanna

doi:10.1016/j.jde.2016.03.016

Cited by 29 publications

(38 citation statements)

References 20 publications

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“…In particular, the techinque used in [9, Section 4] for p = 2 and in [18] for p > 2 can be applied also to annular domains, in this case it provides the existence of at least two monotone solutions, one increasing and one decreasing, cf. also [7,Section 3]. The paper is organized as follows.…”

Section: 2mentioning

confidence: 99%

“…The recent literature has shown that, in presence of homogeneus Neumann boundary conditions, quasilinear equations of the type (1.1) typically admit many positive solutions (in addition to the constant one) and that the set of positive solutions has a rich structure. We quote here the articles [31,32,13,1,2,3,41,40,29,39,9,7,8,33,18,6,20], some of which will be discussed later. Let us illustrate this fact in the semilinear case p = 2, when Ω is a ball and f (s) = s q−1 − s with q > 2.…”

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confidence: 99%

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A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth

Boscaggin¹,

Colasuonno²,

Noris³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Let 1 < p < +∞ and let Ω ⊂ R N be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type −∆pu = f (u), u > 0 in Ω, ∂ν u = 0 on ∂Ω. We suppose that f (0) = f (1) = 0 and that f is negative between the two zeros and positive after. In case Ω is a ball, we also require that f grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behavior (around 1) of non-constant radial solutions.

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Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth

Boscaggin¹,

Colasuonno²,

Noris³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…This nontrivial branch consists of solutions having exactly k oscillations around the constant solution 1. We also refer to [6], [4], [5] for other results about this class of problems. As mentioned above, it was conjectured in [8] that a similar behavior should hold also for a general nonlinearity.…”

Section: 2mentioning

confidence: 99%

Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions

2018

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For 1 < p < ∞, we consider the following problem −∆pu = f (u), u > 0 in Ω, ∂ν u = 0 on ∂Ω,where Ω ⊂ R N is either a ball or an annulus. The nonlinearity f is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity f (s) = −s p−1 + s q−1 for every q > p. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution u ≡ 1. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris, T. Weth, Ann. Inst. H. Poincaré Anal. Non Lináire vol. 29, pp. 573-588 (2012)], that is to say, if p = 2 and f (1) > λ rad k+1 , there exists a radial solution of the problem having exactly k intersections with u ≡ 1 for a large class of nonlinearities.

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“…Another closely related results are contained in [15,16,38] where solutions of the stationary Keller-Segel system concentrating on the boundary of the domain were constructed. Finally we mention [3,4,5] (see also [2]) where solutions with unbounded mass to the same problem were found in the radially symmetric setting.…”

Section: R2mentioning

confidence: 88%

Maximal solution of the Liouville equation in doubly connected domains

Kowalczyk

Pistoia

Vaira

2019

Journal of Functional Analysis

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In this paper we consider the Liouville equation ∆u + λ 2 e u = 0 with Dirichlet boundary conditions in a two dimensional, doubly connected domain Ω. We show that there exists a simple, closed curve γ ⊂ Ω such that for a sequence λn → 0 and a sequence of solutions un it holds un log 1 λn → H, where H is a harmonic function in Ω \ γ and λ 2 n log 1 λn Ω e un dx → 8πc Ω , where c Ω is a constant depending on the conformal class of Ω only. 1991 Mathematics Subject Classification. 35J25, 35J20, 35B33, 35B40.

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Multi-layer radial solutions for a supercritical Neumann problem

Cited by 29 publications

References 20 publications

A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth

A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth

Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions

Maximal solution of the Liouville equation in doubly connected domains

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