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A Nonexistence of Solutions to a Supercritical Problem

Received: 04 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.

DOI 10.11648/j.pamj.20130206.13
Published in Pure and Applied Mathematics Journal (Volume 2, Issue 6, December 2013)
Page(s) 184-190
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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
[1] A. Bahri, Critical point at infinity in some variational problems, Pitman Res. Notes Math. Ser. 182, Longman Sci. Tech. Harlow 1989.
[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.
Author Information
  • 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

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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://download.sciencepg.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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    T1  - A Nonexistence of Solutions to a Supercritical Problem
    AU  - Kamal Ould Bouh
    Y1  - 2014/01/10
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    N1  - https://doi.org/10.11648/j.pamj.20130206.13
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    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.
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