On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator

Chaudhuri, N.; Cîrstea, Florica C.

doi:10.1016/j.crma.2008.12.018

Cited by 20 publications

(17 citation statements)

References 12 publications

(21 reference statements)

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“…For background and motivation of our study, see [22], [11] and [15]. Our findings here incorporate several improvements and extensions over those recently published in [11] on (1.3) with 0 < λ < (N − 2) 2 /4, b = 1 and 1 < q < q * , as well as in [15] on (1.25) with m = 2. We confine our details below to −∞ < λ < (N −2) 2 /4.…”

Section: An Outline Of Our Resultssupporting

confidence: 59%

“…Our Theorems 2.1, 2.2 and 2.4 on (1.3) with −∞ < λ < (N − 2) 2 /4 will refine and generalize the main results in [15] on ∆u = b(x)h(u) in Ω * with N ≥ 3, as well as Theorem 1.1 in [11] on (1.3) with b = 1 and 0 < λ < (N − 2) 2 /4. The method of proof outlined in [11] relies essentially on the fact that every positive solution of (1.3) blows-up at zero for 0 < λ < (N − 2) 2 /4. Since this is not the case when λ ≤ 0, the approach in [11] is not applicable to our problem.…”

Section: Background and Aimssupporting

confidence: 74%

“…In this paper, we treat through a different approach the more general setting of (1.3) for the whole range −∞ < λ ≤ (N − 2) 2 /4 and completely describe the asymptotic behaviour near zero for all positive solutions of (1.3). Specifically, our Theorem 2.4 improves and generalizes the main result in [11] and [14] by establishing a trichotomy of positive solutions for (1.3) under optimal conditions. Our structural assumptions on b and h will rely on regular variation theory as in [15].…”

Section: Background and Aimssupporting

confidence: 62%

“…In this paper we are motivated by [22], [15] and [11] to give a complete classification of the singular solutions for a broader class of nonlinear elliptic equations than (1.2). As a byproduct, we resolve Vázquez-Véron's question when h is regularly varying at ∞ of index greater than 1 (see §1.…”

Section: Background and Aimsmentioning

confidence: 99%

“…The method of proof outlined in [11] relies essentially on the fact that every positive solution of (1.3) blows-up at zero for 0 < λ < (N − 2) 2 /4. Since this is not the case when λ ≤ 0, the approach in [11] is not applicable to our problem. In this paper, we treat through a different approach the more general setting of (1.3) for the whole range −∞ < λ ≤ (N − 2) 2 /4 and completely describe the asymptotic behaviour near zero for all positive solutions of (1.3).…”

Section: Background and Aimsmentioning

confidence: 99%

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2013

Memoirs of the AMS

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Chapter 1. Introduction 1.1. Background and aims 1.2. Our framework 1.3. On Vázquez-Véron's open question 1.4. An outline of our results 1.5. Acknowledgements Chapter 2. Main results 2.1. The subcritical parameter 2.2. The critical parameter Chapter 3. Radial solutions in the power case 3.1. The subcritical parameter 3.2. The critical parameter Chapter 4. Basic ingredients 4.1. A priori estimates 4.2. A Harnack-type inequality 4.3. A regularity lemma Chapter 5. The analysis for the subcritical parameter 5.1. Proof of Theorem 2.1 5.2. Proof of Theorem 2.2 5.3. Proof of Theorem 2.3 5.4. Proof of Theorem 2.4 Chapter 6. The analysis for the critical parameter 6.1. Proof of Theorem 2.5 6.2. Proof of Theorems 2.6 6.3. Proof of Theorem 2.7 Chapter 7. Illustration of our results 7.1. On a prototype model 7.2. In other settings Appendix A. Regular variation theory and related results A.1. Properties of regularly varying functions A.

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Section: An Outline Of Our Resultssupporting

confidence: 59%

Section: Background and Aimssupporting

confidence: 74%

Section: Background and Aimssupporting

confidence: 62%

Section: Background and Aimsmentioning

confidence: 99%

Section: Background and Aimsmentioning

confidence: 99%

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2013

Memoirs of the AMS

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Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

et al. 2012

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We prove the uniform boundedness of all solutions for a general class of Dirichlet anisotropic elliptic problems of the formWe assume that f ∈ L r (Ω) with r > N/p. The feature of this study is the inclusion of a possibly singular gradient-dependent term Ψ(u, ∇u) = N j=1 |u| θ j −2 u |∂ju| q j , where θj > 0 and 0 ≤ qj < pj for 1 ≤ j ≤ N . The existence of such weak solutions is contained in a recent paper by the authors.

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Nonexistence of positive supersolutions to a class of semilinear elliptic equations and systems in an exterior domain

2019

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In this paper, we primarily consider the following semilinear elliptic equationwhere Ω is an exterior domain in R N with N 3, h : Ω × R + → R is a measurable function, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive supersolution to an elliptic system. Keywords Semilinear elliptic problem; Supersolution; Nonexistence MSC(2010) 35J60, 35B53Citation: Huyuan Chen, Rui Peng, Feng Zhou. Nonexistence of positive supersolution to a class of semilinear elliptic equations and systems in an exterior domain.

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On trichotomy of positive singular solutions associated with the Hardy–Sobolev operator

Cited by 20 publications

References 12 publications

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Untitled

Local behaviour of singular solutions for nonlinear elliptic equations in divergence form

Nonexistence of positive supersolutions to a class of semilinear elliptic equations and systems in an exterior domain

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