Multiple positive solutions of the stationary Keller–Segel system

Bonheure, Denis; Casteras, Jean-Baptiste; Noris, Benedetta

doi:10.1007/s00526-017-1163-3

Cited by 22 publications

(23 citation statements)

References 24 publications

(40 reference statements)

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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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Section: 2mentioning

confidence: 99%

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

2018

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“…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%

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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“…Suppose Ω is a domain of m revolution which satisfies (3). Assume that a(t 1 , t 2 , ..., t m ) = a(t 1 , t 2 , ..., t i )…”

Section: Main Results and Symmetry Assumptions On ωmentioning

confidence: 99%

A new variational principle, convexity, and supercritical Neumann problems

Cowan

Moameni

2019

Trans. Amer. Math. Soc.

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Utilizing a new variational principle that allows dealing with problems beyond the usual locally compactness structure, we study problems with a supercritical nonlinearity of the typeTo be more precise, Ω is a bounded domain in R N which satisfies certain symmetry assumptions; Ω is a domain of 'm revolution' (1 ≤ m < N and the case of m = 1 corresponds to radial domains) and where a > 0 satisfies compatible symmetry assumptions along with monotonicity conditions. We find positive nontrivial solutions of (1) in the case of suitable supercritical nonlinearities f by finding critical points of I whereover the closed convex cone K m of nonnegative, symmetric and monotonic functions in H 1 (Ω) where F ′ = f and where F * is the Fenchel dual of F . We mention two important comments: firstly that there is a hidden symmetry in the functional I due to the presence of a convex function and its Fenchel dual that makes it ideal to deal with super-critical problems lacking the necessary compactness requirement. Secondly the energy I is not at all related to the classical Euler-Lagrange energy associated with (1). After we have proven the existence of critical points u of I on K m we then unitize a new abstract variational approach (developed by one of the present authors in [27,29]) to show these critical points in fact satisfy −∆u + u = a(x)f (u). In the particular case of f (u) = |u| p−2 u we show the existence of positive nontrivial solutions beyond the usual Sobolev critical exponent.

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Multiple positive solutions of the stationary Keller–Segel system

Cited by 22 publications

References 24 publications

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

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

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

A new variational principle, convexity, and supercritical Neumann problems

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