The proof of the Lane–Emden conjecture in four space dimensions

Souplet, Philippe

doi:10.1016/j.aim.2009.02.014

Cited by 223 publications

(161 citation statements)

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“…As we have mentioned yet, the nonexistence result of entire stable solution yields immediately (with blow-up and scaling argument as in [13,18]) Corollary 3.1. Assume that Ω is a proper subdomain of R n and u is a classical, positive and stable solution of ∆ 2 u = u q in Ω where 1 < q < ∞ if n ≤ 8; or 1 < q < n n−8 + n if n ≥ 9 with n in Theorem 1.1.…”

Section: Some Applicationsmentioning

confidence: 87%

Liouville theorems for stable solutions of biharmonic problem

Wei

2012

Math. Ann.

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Section: Some Applicationsmentioning

confidence: 87%

Liouville theorems for stable solutions of biharmonic problem

Wei

2012

Math. Ann.

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“…The conjecture has been proved to be true for radial solutions in all dimensions in [18]. The cases of N = 3, 4 for the conjecture in general have been also solved recently in [20] and [24] respectively. The interested reader can refer to the above papers and references therein for detailed descriptions (see also the works [4,22], etc.).…”

Section: Introductionmentioning

confidence: 90%

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Wang

Zhu

2012

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Poláčik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.

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“…Zou [24] and further developed by Souplet [28], which is based on a combination of Pohozaev identity, Sobolev inequality on the unit sphere S N −1 and measure and feedback arguments. However, we point out that some additional difficulties arise in our problem.…”

Section: Uses the Technique Introduced By Serrin Andmentioning

confidence: 99%

Liouville-type theorems for a quasilinear elliptic equation of the Hénon-type

Phan

Duong

2015

Nonlinear Differ. Equ. Appl.

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We consider the Hénon-type quasilinear elliptic equation −Δmu = |x| a u p where Δmu = div(|∇u| m−2 ∇u), m > 1, p > m − 1 and a ≥ 0. We are concerned with the Liouville property, i.e. the nonexistence of positive solutions in the whole space R N . We prove the optimal Liouville-type theorem for dimension N < m + 1 and give partial results for higher dimensions.Mathematics Subject Classification. Primary 35B53, 35J62; Secondary 35K57, 35B33.

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The proof of the Lane–Emden conjecture in four space dimensions

Cited by 223 publications

References 20 publications

Liouville theorems for stable solutions of biharmonic problem

Liouville theorems for stable solutions of biharmonic problem

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems for a quasilinear elliptic equation of the Hénon-type

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