Large energy entire solutions for the Yamabe equation

Manuel Del Pino; Monica Musso; Frank Pacard; Angela Pistoia

doi:10.1016/j.jde.2011.03.008

Large energy entire solutions for the Yamabe equation

Manuel Del Pino, Monica Musso, Frank Pacard, Angela Pistoia

Department of Mathematical Sciences

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

85 Citations (SciVal)

Abstract

We consider the Yamabe equation Δu+n(n-2)/4|u|^4/n-2u=0 in R{double-struck}ⁿ, n≥3. Let k≥1 and ξ_j ^k=(e^2jπi/k,0)εR{double-struck}ⁿ=C{double-struck}×R{double-struck}^n-2. For all large k we find a solution of the form u_k(x)=U(x)-σ_j=1 ^k;μk-^n-2/2U×(μ_k ^-1(x-ξ_j))+o(1), where U(x)=(2/1+|x|²)^n-2/2, μ_k=c_n/k² for n≥4, μ_k=c/k²(logk)² for n=3 and o(1)→0 uniformly as k→+∞.

Original language	English
Pages (from-to)	2568-2597
Number of pages	30
Journal	Journal of Differential Equations
Volume	251
Issue number	9
DOIs	https://doi.org/10.1016/j.jde.2011.03.008
Publication status	Published - 1 Nov 2011

Funding

The first author is partially supported by Fondecyt Grant 1070389, Fondo Basal CMM and CAPDE-Anillo ACT-125. The second author is partially supported by Fondecyt Grant 1080099 and CAPDE-Anillo ACT-125. The third author is partially supported by the ANR-08-BLANC-0335-01 grant. The last author is supported by the Mi.U.R. National Project Metodi variazionali e topologici nello studio di fenomeni non lineari.

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jde.2011.03.008

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title = "Large energy entire solutions for the Yamabe equation",

abstract = "We consider the Yamabe equation Δu+n(n-2)/4|u|4/n-2u=0 in R\{double-struck\}n, n≥3. Let k≥1 and ξj k=(e2jπi/k,0)εR\{double-struck\}n=C\{double-struck\}×R\{double-struck\}n-2. For all large k we find a solution of the form uk(x)=U(x)-σj=1 k;μk-n-2/2U×(μk -1(x-ξj))+o(1), where U(x)=(2/1+|x|2)n-2/2, μk=cn/k2 for n≥4, μk=c/k2(logk)2 for n=3 and o(1)→0 uniformly as k→+∞.",

author = "\{Del Pino\}, Manuel and Monica Musso and Frank Pacard and Angela Pistoia",

year = "2011",

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doi = "10.1016/j.jde.2011.03.008",

language = "English",

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T1 - Large energy entire solutions for the Yamabe equation

AU - Del Pino, Manuel

AU - Musso, Monica

AU - Pacard, Frank

AU - Pistoia, Angela

PY - 2011/11/1

Y1 - 2011/11/1

N2 - We consider the Yamabe equation Δu+n(n-2)/4|u|4/n-2u=0 in R{double-struck}n, n≥3. Let k≥1 and ξj k=(e2jπi/k,0)εR{double-struck}n=C{double-struck}×R{double-struck}n-2. For all large k we find a solution of the form uk(x)=U(x)-σj=1 k;μk-n-2/2U×(μk -1(x-ξj))+o(1), where U(x)=(2/1+|x|2)n-2/2, μk=cn/k2 for n≥4, μk=c/k2(logk)2 for n=3 and o(1)→0 uniformly as k→+∞.

AB - We consider the Yamabe equation Δu+n(n-2)/4|u|4/n-2u=0 in R{double-struck}n, n≥3. Let k≥1 and ξj k=(e2jπi/k,0)εR{double-struck}n=C{double-struck}×R{double-struck}n-2. For all large k we find a solution of the form uk(x)=U(x)-σj=1 k;μk-n-2/2U×(μk -1(x-ξj))+o(1), where U(x)=(2/1+|x|2)n-2/2, μk=cn/k2 for n≥4, μk=c/k2(logk)2 for n=3 and o(1)→0 uniformly as k→+∞.

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DO - 10.1016/j.jde.2011.03.008

M3 - Article

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SN - 0022-0396

VL - 251

SP - 2568

EP - 2597

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 9

ER -

Large energy entire solutions for the Yamabe equation

Abstract

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