Existence and Multiplicity of Nodal Solutions for Nonlinear Elliptic Equations with Critical Sobolev Growth

Hebey, Emmanuel; Vaugon, Michel

doi:10.1006/jfan.1994.1012

Cited by 55 publications

(35 citation statements)

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“…Si λ p,γ = 0 alors w p,γ = 0 , ce qui provient de l'équation (10) et de la condition (8). On en déduit une contradiction avec le fait que w p,γ ∈ A p,γ .…”

Section: Convergence Vers Une Solutionunclassified

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Solutions nodales sur les variétés Riemanniennes non localement conformément plates à bord

Holcman

2001

Comment. Math. Helv.

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Abstract. Dans cet article, on considère le problème de Dirichlet pour une EDP non linéaire avec une donnée au bord qui change de signe. L'existence d'une famille de solutions minimisantes est obtenue sur les variétés riemaniennes compactes non localement conformément platesà bord. This paper deals with solutions of Dirichlet problem for a nonlinear PDE with boundary data changing of sign. A family of minimizing nodal solutions are found on nonlocally conformally flat manifolds with boundary. Mathematics Subject Classification (2000). 53C21, 58J05.

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“…Si λ p,γ = 0 alors w p,γ = 0 , ce qui provient de l'équation (10) et de la condition (8). On en déduit une contradiction avec le fait que w p,γ ∈ A p,γ .…”

Section: Convergence Vers Une Solutionunclassified

“…Rappelons qu'une solution non minimisante aété obtenue par une analysé elémentaire lorsque la donnée au bord φ n'est pas identiquement nulle: il aété montré [8] que pour tout > 0 , il existe un réel λ ∈ (0, ) et une fonction u de classe C 2 tels que…”

Section: Introductionunclassified

Solutions nodales sur les variétés Riemanniennes non localement conformément plates à bord

Holcman

2001

Comment. Math. Helv.

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“…From there we infer that H has only finitely many negative eigenvalues, whose eigenfunctions span a vector space of dimensionm; as (Φ 0 ℓ ) ℓ=1,...,l span ker H, and this yields (24). Also notice that if we denoteṼ (x) := V (Λ β (x)) then (LṼ )(x) = (LV )(Λ β (x)).…”

Section: Moreover There Existsmentioning

confidence: 96%

“…Here are some references on the construction of such solutions. Hebey-Vaugon [24] showed the existence of nodal (i.e sign changing) solutions (see also the references therein for radial solutions) in dimension d 3. More recently, Del Pino, Musso, Pacard and Pistoia constructed in [12] solution to the massless version of equation (7) with a centered soliton crowned with negative spikes (rescaled solitons) at the vertices of a regular polygon of radius 1; in [13], they constructed sign changing, non radial solutions to (7) on the sphere S d (d 4) whose energy is concentrated along special submanifolds of S d .…”

Section: Introductionmentioning

confidence: 99%

Multi-travelling waves for the nonlinear Klein-Gordon equation

Côte¹,

Martel²

2018

Trans. Amer. Math. Soc.

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“…In particular, if blow-up occurs, then we get with (1.16) (see also Collion [4] and Hebey [13]) that…”

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confidence: 93%

Sharp Sobolev inequalities in the presence of a twist

Collion¹,

Hebey²,

Vaugon³

2007

Trans. Amer. Math. Soc.

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Abstract. Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Let also A be a smooth symmetrical positive (0, 2)-tensor field in M . By the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where H 2 1 (M ) is the standard Sobolev space of functions in L 2 with one derivative in L 2 . We investigate in this paper the value of the sharp K in the equation above, the validity of the corresponding sharp inequality, and the existence of extremal functions for the saturated version of the sharp inequality.Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Also let A be a smooth symmetrical (0, 2)-tensor field in M . In a local chart, A = (A ij ), i, j = 1, . . . , n. We assume that A is positive when acting on 1-forms in the sense that for any x ∈ M , and any η in the cotangent spaceThen, by the Sobolev embedding theorem, we can write that there exist K, B > 0 such that for any u ∈ H 2 1 (M ),where ∇u = (∂ i u) is the 1-form consisting (in local charts) of the first derivatives of u, dv g is the Riemannian volume element of g, and H 2 1 (M ) is the standard Sobolev space consisting of functions in L 2 with one derivative in L 2 . Clearly, the sharp constant B in (0.1) is V −2/n g , where V g is the volume of (M, g), and the corresponding sharp inequality holds true since it holds true for the classical Sobolev inequality [and |∇u| 2 is controled by A x (∇u, ∇u)]. On the other hand, as is easily understood by the fact that A charges some parts of the space M more than others, it is expected that A will affect the sharp constant K in (0.1). Note (0.1) is associated to the operator ∆ g A u = −div g (A x ∇u) which appears in several places in mathematical and physics literature.The questions we ask in this note are: what is the value K s = K s (g) of the sharp K in (0.1), does the corresponding sharp inequality hold true, and if yes, does its saturated version (where B is lowered to its minimum value under the constraint K = K s ) possess extremal functions? When A = g −1 , we are back to the classical problem (dealing with the classical Sobolev inequality). Possible references in book

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Existence and Multiplicity of Nodal Solutions for Nonlinear Elliptic Equations with Critical Sobolev Growth

Cited by 55 publications

References 0 publications

Solutions nodales sur les variétés Riemanniennes non localement conformément plates à bord

Solutions nodales sur les variétés Riemanniennes non localement conformément plates à bord

Multi-travelling waves for the nonlinear Klein-Gordon equation

Sharp Sobolev inequalities in the presence of a twist

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