Skip to main content
SpringerLink
Log in

Construction of solutions of an supercritical elliptic PDE in low dimensions

  • Published:
Annali di Matematica Pura ed Applicata (1923 -) Aims and scope Submit manuscript

Abstract

In this paper, we study the supercritical problem \((P_\varepsilon )\): \( -\triangle u=K|u|^{4/(n-2)+\varepsilon }u~ \text{ in } \Omega , ~ u=0 \text{ on } \partial \Omega ,\) where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^n\), \(n=3,4\), \(\varepsilon \) is a positive real parameter and K is a \(C^4\) positive function on \({\overline{\Omega }}\). Following the ideas of Bahri et al. (Calc Var Partial Differ Equ 8:67–93, 1995) and using the finite-dimensional reduction, we construct some solutions of \((P_\varepsilon )\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosetti, A., Azorero, G., Peral, J.: Perturbation of \(\Delta u + u^{(N+2)/(N-2)} = 0\), the scalar curvature problem in \({\mathbb{R}}^N\), and related topics. J. Funct. Anal. 165, 117–149 (1999)

    Article  MathSciNet  Google Scholar 

  2. Ben Ayed, M.: Finite dimensional reduction of a supercritical exponent equation. Tunis. J. Math. 2(2), 379–397 (2020)

    Article  MathSciNet  Google Scholar 

  3. Ben Ayed, M., Ould Bouh, K.: Nonexistence results of sign-changing solutions to supercritical nonlinear problem. Commun. Pure Appl. Anal. 7(5), 1057–1075 (2008)

    Article  MathSciNet  Google Scholar 

  4. Ben Ayed, M., Ould Bouh, K.: Construction of solutions of an elliptic PDE with a supercritical exponent nonlinearity using the finite dimensional reduction. Monatshefte f\(\ddot{u}\)r Mathematik 192, 49–63 (2020). https://doi.org/10.1007/s00605-020-01386-8

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben Ayed, M., Ghoudi, R.: Profile and existence of sign-changing solutions to an elliptic subcritical equations. Commun. Contemp. Math. 10, 1183–1216 (2008)

    Article  MathSciNet  Google Scholar 

  6. Ben Ayed, M., EL Mehdi, K., Grossi, M., Rey, O.: A nonexistence results of single peaked solutions to a supercritical nonlinear problem. Commun. Contemp. Math. 2, 179–195 (2003)

    Article  MathSciNet  Google Scholar 

  7. Bahri, A.: Critical Point at Infinity in Some Variational Problem. Pitman Res. Notes Math, Ser 182. Longman Sci. Tech. Harlow (1989)

  8. Bahri, A., Coron, J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of topology of the domain. Commun. Pure Appl. Math. 41, 255–294 (1988)

    MathSciNet  MATH  Google Scholar 

  9. Bahri, A., Coron, J.M.: The scalar curvature problem on the standard three dimensional spheres. J. Funct. Anal. 95, 106–172 (1991)

    Article  MathSciNet  Google Scholar 

  10. Bahri, A., Li, Y.Y., Rey, O.: On a variational problem with lack of compactness: the topological effect of the critical points at infinity. Calc. Var. Partial Differ. Equ. 3, 67–93 (1995)

    Article  MathSciNet  Google Scholar 

  11. Bartsch, T.: Critical point theory on partially ordered Hilbert spaces. J. Funct. Anal. 186, 117–152 (2001)

    Article  MathSciNet  Google Scholar 

  12. Bartsch, T., Weth, T.: A note on additional properties of sign-changing solutions to super-linear elliptic equations. Topol. Methods Nonlinear Anal. 22, 1–14 (2003)

    Article  MathSciNet  Google Scholar 

  13. Cao, D., Noussair, E., Yan, S.: On the scalar curvature equation \(-\Delta u = (1+\varepsilon K)u^{(N+2)/(N-2)}\) in \({\mathbb{R}}^N\). Calc. Var. Partial Differ. Equ. 15, 403–419 (2002)

    Article  Google Scholar 

  14. Castro, A., Cossio, J., Newgerger, J.M.: A sign-changing solution for a supercritical Dirichlet problem. Rocky Mt. J. Math. 27, 1041–1053 (1997)

    Article  Google Scholar 

  15. Clapp, M., Weth, T.: Minimal nodal solutions of the pure critical exponent problem on a symmetric domain. Calc. Var. Partial Differ. Equ. 21, 1–14 (2004)

    Article  MathSciNet  Google Scholar 

  16. Coron, J.M.: Topologie et cas limite des injections de Sobolev. C. R. Acad. Sci. Paris Ser. I 299, 209–212 (1984)

    MathSciNet  MATH  Google Scholar 

  17. Dammak, Y., Ghoudi, R.: Sign-changing tower of bubbles to an elliptic subcritical equation. Commun. Contemp. Math. 21(7), 1850052 (2019). https://doi.org/10.1142/S0219199718500529

    Article  MathSciNet  MATH  Google Scholar 

  18. Dancer, E.N.: A note on an equation with critical exponent. Bull. Lond. Math. Soc. 20, 600–602 (1988)

    Article  MathSciNet  Google Scholar 

  19. Del Pino, M., Felmer, P., Musso, M.: Multi-peak solutions for supercritical elliptic problems in domains with small holes. J. Differ. Equ. 182, 511–540 (2002)

    Article  Google Scholar 

  20. Del Pino, M., Felmer, P., Musso, M.: Two bubbles solutions in the supercritical Bahri–Coron’s problem. Calc. Var. Partial Differ. Equ. 16, 113–145 (2003)

    Article  Google Scholar 

  21. Ding, W.Y.: Positive solution of \(\Delta u+u^{(n+2)/(n-2)}=0\) on contractible domain. J. Partial Differ. Equ. 2, 83–88 (1988)

    MathSciNet  MATH  Google Scholar 

  22. Ghoudi, R.: On a variational problem involving critical Sobolev growth in dimension three. Nonlinear Differ. Equ. Appl. 23, 2 (2016). https://doi.org/10.1007/s00030-016-0360-7

    Article  MathSciNet  MATH  Google Scholar 

  23. Ghoudi, R., Bouh, K.: Construction of sign-changing solutions for a subcritical problem on the four dimensional half sphere. Tohoku Math. J. 68, 591–605 (2016)

    Article  MathSciNet  Google Scholar 

  24. Hebey, E.: La méthode de isométrie-concentration dans le cas d’un problème non linéaire sur les variétés compactes à bord. Bull. Sci. Math. 116, 35–51 (1992)

    MathSciNet  MATH  Google Scholar 

  25. Khenissi, S., Rey, O.: A criterion for existence of solutions to the supercritical Bahri–Coron’s problem. Houst. J. Math. 30, 587–613 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Li, Y.Y.: Prescribing scalar curvature on \(S^n\) and related problems, part I. J. Differ. Equ. 120, 319–410 (1995)

    Article  Google Scholar 

  27. Li, Y.Y.: Prescribing scalar curvature on \(S^n\) and related problems, part II: existence and compactness. Commun. Pure Appl. Math. 49, 541–597 (1996)

    Article  Google Scholar 

  28. Micheletti, A.M., Pistoia, A., Vétois, J.: Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds. Indiana Univ. Math. J. 58, 1719–1746 (2009)

    Article  MathSciNet  Google Scholar 

  29. Musso, M., Salazar, D.: Multispike solutions for the Brezis–Nirenberg problem in dimension three. J. Differ. Equ. 264(11), 6663–6709 (2018)

    Article  MathSciNet  Google Scholar 

  30. Passaseo, D.: Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. Manuscr. Math. 65(2), 147–165 (1989)

    Article  MathSciNet  Google Scholar 

  31. Passaseo, D.: Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains. J. Funct. Anal. 114(1), 97–105 (1993)

    Article  MathSciNet  Google Scholar 

  32. Passaseo, D.: Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains. Duke Math. J. 92(2), 429–457 (1998)

    Article  MathSciNet  Google Scholar 

  33. Passaseo, D.: The effect of the domain shape on the existence of positive solutions of the equation \(\triangle u+u^{2^*-1}= 0\). Topol. Methods Nonlinear Anal. 3, 27–54 (1994)

    Article  MathSciNet  Google Scholar 

  34. Pistoia, A., Weth, T.: Sign-changing bubble-tower solutions in a slightly subcritical semi-linear Dirichlet problem. Ann. Inst. H. Poincaré Anal. Nonlinéaire 24, 325–340 (2007)

    Article  Google Scholar 

  35. Rey, O.: The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent. J. Funct. Anal. 89, 1–52 (1990)

    Article  MathSciNet  Google Scholar 

  36. Rey, O.: The topological impact of critical points at infinity in a variational problem with lack of compactness: the dimension 3. Adv. Differ. Equ. 4, 581–616 (1999)

    MathSciNet  MATH  Google Scholar 

  37. Wei, J., Yan, S.: Infinitely many solutions for prescribed curvature problem on \(S^n\). J. Funct. Anal. 258, 3038–3081 (2010)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculté des Sciences de Sfax, Université de Sfax, BP 1171, Route Soukra, 3000, Sfax, Tunisia

    Mohamed Ben Ayed

  2. Ecole Supérieure de Commerce, Université de Sfax, BP 1081, Route de l’aéroport, 3018, Sfax, Tunisia

    Yessine Dammak

Authors
  1. Mohamed Ben Ayed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yessine Dammak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yessine Dammak.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ben Ayed, M., Dammak, Y. Construction of solutions of an supercritical elliptic PDE in low dimensions. Annali di Matematica 200, 51–66 (2021). https://doi.org/10.1007/s10231-020-00982-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10231-020-00982-7

Keywords

Mathematics Subject Classification

Search

Navigation