Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3

Manuel del Pino; Monica Musso; Carlos Román; Juncheng Wei

doi:10.1007/s11854-019-0008-8

Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3

Manuel del Pino, Monica Musso, Carlos Román, Juncheng Wei

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

7 Citations (SciVal)

Abstract

We consider the problem of finding positive solutions of the problem Δu − λu + u ⁵ = 0 in a bounded, smooth domain Ω in ℝ ³ , under zero Neumann boundary conditions. Here λ is a positive number. We analyze the role of Green’s function of −Δ + λ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when λ is regarded as a parameter. As a special case of our results, we find and characterize a positive value λ _* such that if λ − λ _* > 0 is sufficiently small, then this problem is solvable by a solution u _λ which blows-up by bubbling at a certain interior point of Ω as λ ↓ λ _* .

Original language	English
Pages (from-to)	813-843
Number of pages	31
Journal	Journal d'Analyse Mathematique
Volume	137
Issue number	2
DOIs	https://doi.org/10.1007/s11854-019-0008-8
Publication status	Published - 1 Mar 2019

Funding

where θλ is as in Lemma 4.1. Thus, ψλ has a critical point as in the statement of Lemma 4.1. This concludes the proof of our main result, with the constant γ = 2a2a1. □ Acknowledgement. The research of the first author has been suported by grants Fondecyt 1150066 and Fondo Basal CMM. The second author has been supported by Millenium Nucleus CAPDE, NC130017 and Fondecyt grant 1160135. The third author has been supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098, LabEx SMP).

ASJC Scopus subject areas

Analysis
General Mathematics

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10.1007/s11854-019-0008-8

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title = "Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3",

abstract = " We consider the problem of finding positive solutions of the problem Δu − λu + u 5 = 0 in a bounded, smooth domain Ω in ℝ 3 , under zero Neumann boundary conditions. Here λ is a positive number. We analyze the role of Green{\textquoteright}s function of −Δ + λ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when λ is regarded as a parameter. As a special case of our results, we find and characterize a positive value λ * such that if λ − λ * > 0 is sufficiently small, then this problem is solvable by a solution u λ which blows-up by bubbling at a certain interior point of Ω as λ ↓ λ * . ",

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AU - del Pino, Manuel

AU - Musso, Monica

AU - Román, Carlos

AU - Wei, Juncheng

PY - 2019/3/1

Y1 - 2019/3/1

N2 - We consider the problem of finding positive solutions of the problem Δu − λu + u 5 = 0 in a bounded, smooth domain Ω in ℝ 3 , under zero Neumann boundary conditions. Here λ is a positive number. We analyze the role of Green’s function of −Δ + λ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when λ is regarded as a parameter. As a special case of our results, we find and characterize a positive value λ * such that if λ − λ * > 0 is sufficiently small, then this problem is solvable by a solution u λ which blows-up by bubbling at a certain interior point of Ω as λ ↓ λ * .

AB - We consider the problem of finding positive solutions of the problem Δu − λu + u 5 = 0 in a bounded, smooth domain Ω in ℝ 3 , under zero Neumann boundary conditions. Here λ is a positive number. We analyze the role of Green’s function of −Δ + λ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when λ is regarded as a parameter. As a special case of our results, we find and characterize a positive value λ * such that if λ − λ * > 0 is sufficiently small, then this problem is solvable by a solution u λ which blows-up by bubbling at a certain interior point of Ω as λ ↓ λ * .

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JF - Journal d'Analyse Mathematique

IS - 2

ER -

Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3

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Concentration phenomena in nonlinear analysis

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Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3

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Concentration phenomena in nonlinear analysis

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