On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain

Bahri, Abbas; Coron, Jean‐Michel

doi:10.1002/cpa.3160410302

Cited by 642 publications

(713 citation statements)

References 16 publications

(8 reference statements)

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“…For the second order problem −∆u = |u| 4/(n−2) u in Ω, u = 0 on ∂Ω, it was shown with help of very subtle methods by Coron [7] and Bahri-Coron [2] that this result extends to domains with nontrivial topology. Recently, corresponding results have been found also in the higher order case: see [11] for problem (5) and [3] for the polyharmonic analogue of (6) under Dirichlet boundary conditions.…”

Section: Theoremmentioning

confidence: 99%

Existence and nonexistence results for critical growth biharmonic elliptic equations

Gazzola

Grunau

Squassina

2003

Calculus of Variations and Partial Differential Equations

123

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Abstract. We prove existence of nontrivial solutions to semilinear fourth order problems at critical growth in some contractible domains which are perturbations of small capacity of domains having nontrivial topology. Compared with the second order case, some difficulties arise which are overcome by a decomposition method with respect to pairs of dual cones. In the case of Navier boundary conditions, further technical problems have to be solved by means of a careful application of concentration compactness lemmas. The required generalization of a Struwe type compactness lemma needs a somehow involved discussion of certain limit procedures.Also nonexistence results for positive solutions in the ball are obtained, extending a result of Pucci and Serrin on so-called critical dimensions to Navier boundary conditions. A Sobolev inequality with optimal constant and remainder term is proved, which is closely related to the critical dimension phenomenon. Here, this inequality serves as a tool in the proof of the existence results and in particular in the discussion of certain relevant energy levels.

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Section: Theoremmentioning

confidence: 99%

Existence and nonexistence results for critical growth biharmonic elliptic equations

Gazzola

Grunau

Squassina

2003

Calculus of Variations and Partial Differential Equations

123

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“…Since the work by Bahri and Coron [2], the effect of the domain topology on the existence and multiplicity of the solutions is one of the subjects which attract much attention. See, for example, [2,3,4,10,11,12,14,15].…”

Section: For Any Set M Denotementioning

confidence: 99%

The effect of the graph topology on the existence ofmultipeak solutions for nonlinear Schrödinger equations

Dancer¹,

Lam²,

Yan³

2001

Abstract and Applied Analysis

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“…In case n − ≥ 3, p − = 2, and p + = 2 * , see Bahri-Coron [9], we get the existence of a nonnegative, nontrivial solution of problem (2.1) with ε = 0 for noncontractible, smooth, bounded domains Ω 1 . Therefore, we state a second corollary of Proposition 2.1 as follows.…”

Section: Examples Of Solutions and Palais-smale Sequencesmentioning

confidence: 96%

The blow-up of critical anisotropic equations with critical directions

Vétois

2010

Nonlinear Differ. Equ. Appl.

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On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain

Cited by 642 publications

References 16 publications

Existence and nonexistence results for critical growth biharmonic elliptic equations

Existence and nonexistence results for critical growth biharmonic elliptic equations

The effect of the graph topology on the existence ofmultipeak solutions for nonlinear Schrödinger equations

The blow-up of critical anisotropic equations with critical directions

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