A Nonexistence of Solutions to a Supercritical Problem

Kamal Ould Bouh

doi:doi:10.11648/j.pamj.20130206.13

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A Nonexistence of Solutions to a Supercritical Problem

Kamal Ould Bouh

Published in Pure and Applied Mathematics Journal (Volume 2, Issue 6)

Received: 4 December 2013 Accepted: Published: 10 January 2014

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Abstract

In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (P_ϵ ) ∶ -∆u=K u^(□((n+2)/(n-2))+ϵ) in Ω ; u >0 in Ω and u=0 on ∂ Ω where is a smooth bounded domain in 〖IR〗^n n≥3, K is a C^3positive function and ϵ is a small positive real parameter. We prove that, for small, (Pε) has no positive solutions which blow up at one critical point of the function K.

Published in	Pure and Applied Mathematics Journal (Volume 2, Issue 6)
DOI	10.11648/j.pamj.20130206.13
Page(s)	184-190
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Nonlinear Elliptic Equations, Critical Exponent, Variational Problem

References

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[2]	A. Bahri and J. M. Coron, On a nonlinear elliptic equation involving the Sobolev exponent : the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294
[3]	A. Bahri, Y.Y. Li and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var. and Part. Diff. Equ. 3 (1995), 67-94.
[4]	M. Ben Ayed, K. El Mehdi, M. Grossi and O. Rey, A Nonexistence result of single peaked solutions to a supercritical nonlinear problem, Comm. Contenporary Math., 2 (2003), 179-195.
[5]	M. Ben Ayed, K. Ould Bouh, Nonexistence results of sign-changing solutions to a supercritical nonlinear problem, Comm. Pure Applied Anal, 5 (2007), 1057-1075.
[6]	M. Del Pino, P. Felmer and M. Musso, Two bubles solutions in the supercritical Bahri-Coron’s problem, Calc. Var. Part. Diff. Equat., 16 (2003), 113–145.
[7]	Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincare (Analyse non-linear) 8(1991),159-174.
[8]	K. Ould Bouh, Nonexistence result of sign-changing solutions for a supercritical problem of the scalar curvature type , Advance in Nonlinear Studies (ANS), 12 (2012), 149-171.
[9]	O. Rey, The role of Green’s function in a nonlinear elliptic equation involving critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1-52.
[10]	O. Rey, The topological impact of critical points at infinity in a variational problem with lack of compactness : the dimension 3, Adv. Diff. Equ. 4 (1999), 581-616.

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APA Style

Kamal Ould Bouh. (2014). A Nonexistence of Solutions to a Supercritical Problem. Pure and Applied Mathematics Journal, 2(6), 184-190. https://doi.org/10.11648/j.pamj.20130206.13

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Kamal Ould Bouh. A Nonexistence of Solutions to a Supercritical Problem. Pure Appl. Math. J. 2014, 2(6), 184-190. doi: 10.11648/j.pamj.20130206.13

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AMA Style

Kamal Ould Bouh. A Nonexistence of Solutions to a Supercritical Problem. Pure Appl Math J. 2014;2(6):184-190. doi: 10.11648/j.pamj.20130206.13

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@article{10.11648/j.pamj.20130206.13,
  author = {Kamal Ould Bouh},
  title = {A Nonexistence of Solutions to a Supercritical Problem},
  journal = {Pure and Applied Mathematics Journal},
  volume = {2},
  number = {6},
  pages = {184-190},
  doi = {10.11648/j.pamj.20130206.13},
  url = {https://doi.org/10.11648/j.pamj.20130206.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130206.13},
  abstract = {In this paper, we study the nonlinear elliptic problem involving nearly critical exponent  (P_ϵ ) ∶ -∆u=K u^(□((n+2)/(n-2))+ϵ)    in  Ω  ; u >0   in   Ω  and  u=0  on ∂ Ω   where   is a smooth bounded domain in 〖IR〗^n n≥3, K is a C^3positive function and ϵ is a small positive real parameter. We prove that, for   small, (Pε) has no positive solutions which blow up at one critical point of the function K.},
 year = {2014}
}

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TY  - JOUR
T1  - A Nonexistence of Solutions to a Supercritical Problem
AU  - Kamal Ould Bouh
Y1  - 2014/01/10
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20130206.13
DO  - 10.11648/j.pamj.20130206.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 184
EP  - 190
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130206.13
AB  - In this paper, we study the nonlinear elliptic problem involving nearly critical exponent  (P_ϵ ) ∶ -∆u=K u^(□((n+2)/(n-2))+ϵ)    in  Ω  ; u >0   in   Ω  and  u=0  on ∂ Ω   where   is a smooth bounded domain in 〖IR〗^n n≥3, K is a C^3positive function and ϵ is a small positive real parameter. We prove that, for   small, (Pε) has no positive solutions which blow up at one critical point of the function K.
VL  - 2
IS  - 6
ER  -

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Author Information

Kamal Ould Bouh

Department of Mathematics, Taibah University, Almadinah Almunawwarah, KSA

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APA Style

Kamal Ould Bouh. (2014). A Nonexistence of Solutions to a Supercritical Problem. Pure and Applied Mathematics Journal, 2(6), 184-190. https://doi.org/10.11648/j.pamj.20130206.13

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ACS Style

Kamal Ould Bouh. A Nonexistence of Solutions to a Supercritical Problem. Pure Appl. Math. J. 2014, 2(6), 184-190. doi: 10.11648/j.pamj.20130206.13

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AMA Style

Kamal Ould Bouh. A Nonexistence of Solutions to a Supercritical Problem. Pure Appl Math J. 2014;2(6):184-190. doi: 10.11648/j.pamj.20130206.13

Copy | Download

@article{10.11648/j.pamj.20130206.13,
  author = {Kamal Ould Bouh},
  title = {A Nonexistence of Solutions to a Supercritical Problem},
  journal = {Pure and Applied Mathematics Journal},
  volume = {2},
  number = {6},
  pages = {184-190},
  doi = {10.11648/j.pamj.20130206.13},
  url = {https://doi.org/10.11648/j.pamj.20130206.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20130206.13},
  abstract = {In this paper, we study the nonlinear elliptic problem involving nearly critical exponent  (P_ϵ ) ∶ -∆u=K u^(□((n+2)/(n-2))+ϵ)    in  Ω  ; u >0   in   Ω  and  u=0  on ∂ Ω   where   is a smooth bounded domain in 〖IR〗^n n≥3, K is a C^3positive function and ϵ is a small positive real parameter. We prove that, for   small, (Pε) has no positive solutions which blow up at one critical point of the function K.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - A Nonexistence of Solutions to a Supercritical Problem
AU  - Kamal Ould Bouh
Y1  - 2014/01/10
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20130206.13
DO  - 10.11648/j.pamj.20130206.13
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 184
EP  - 190
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20130206.13
AB  - In this paper, we study the nonlinear elliptic problem involving nearly critical exponent  (P_ϵ ) ∶ -∆u=K u^(□((n+2)/(n-2))+ϵ)    in  Ω  ; u >0   in   Ω  and  u=0  on ∂ Ω   where   is a smooth bounded domain in 〖IR〗^n n≥3, K is a C^3positive function and ϵ is a small positive real parameter. We prove that, for   small, (Pε) has no positive solutions which blow up at one critical point of the function K.
VL  - 2
IS  - 6
ER  -

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