Results in Mathematics

, 74:69 | Cite as

\(L^\infty \)-Norm Estimates of Weak Solutions via Their Morse Indices for the m-Laplacian Problems

  • Mohamed Karim HamdaniEmail author
  • Abdellaziz Harrabi
Open Access


This work is devoted to obtain the \(L^p\) and the \(L^{\infty }\)-estimates of solutions via their Morse indices to the following m-Laplacian problems
$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _m u= f(x,u) \quad \text{ in }\quad \Omega \\ u=0, \quad \text{ on } \partial \Omega , \end{array}\right. }\qquad \qquad \qquad (1) \end{aligned}$$
where \(\Omega \subset {\mathbf {R}}^N\) is a bounded domain with smooth boundary, \(N>m>2\) and \(f\in C(\overline{\Omega }\times {\mathbb {R}})\) which will be specified later. As far as we know, it seems to be the first time that such explicit estimates are obtained for a nonlinear degenerate problems. So, our main results extend and complement previously \(L^{\infty }\)-estimates results in the literature.


Morse index elliptic estimates Pohozaev identity m-Laplacian operator 

Mathematics Subject Classification

Primary 35J30 Secondary 35B38 35J35 35J40 58E30 



The authors would like to express their appreciation to the anonymous referees for useful comments and valuable suggestions which help us in depth to improve the presentation of paper. The first author would like to express his deepest gratitude to the Military School of Aeronautical Specialties, Sfax (ESA) for providing us with an excellent atmosphere for doing this work. This work was partially done while the second author was visited the I.C.T.P, in July 2017.


  1. 1.
    Brezis, H., Kato, T.: Remarks on the Schrödinger operator with singular compact potentials. J. Math. Pures Appl. 58, 137–151 (1978)zbMATHGoogle Scholar
  2. 2.
    Bahri, A., Lions, P.L.: Solutions of superlinear elliptic equations and their Morse indices. Commun. Pure Appl. Math. 45, 1205–1215 (1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dibenedetto, E.: \(C^{1+\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827–850 (1983)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Damascelli, L., Farina, A., Sciunzi, B., Valdinoci, E.: Liouville results for \(m\)-Laplace equations of Lane Emden Fowler type. Ann. I. H. Poincaré. 26, 1099–1119 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Damascelli, L., Sciunzi, B.: Regularity, monotonicity and symmetry of positive solutions of \(m\)-Laplace, equations. J. Differ. Equ. 206(2), 483–515 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Damascelli, L., Sciunzi, B.: Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of \(m\)-Laplace equations. Calc. Var. Partial Differ. Equ. 25(2), 139–159 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    de Figueiredo, D.G., Lions, P.L., Nussbaum, R.: A priori estimates and existence of positive solutions of semilinear elliptic equations. J. Math. Pures Appl. 61, 41–63 (1982)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Farina, A.: On the classification of solutions of the Lane-Emden equation on unbounded domains of \({\mathbb{R}}^N\). J. Math. Pures Appl. 87, 537–561 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Partial Differ. Equ. 6(8), 883–901 (1981)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hajlaoui, H., Harrabi, A., Mtiri, F.: Morse indices of solutions for super-linear elliptic PDE’s. Nonlinear Anal. 116, 180–192 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Harrabi, A.: High-order Bahri-Lions Liouville type Theorems. Annali di Matematica Pura ed Applicata (2016)Google Scholar
  12. 12.
    Kazaniecki, K., Lasica, M., Mazowiecka, K.E., Strzelzcki, P.: A conditional regularity result for \(p\)-Harmonic flows. arXiv:1406.1978v5 (2015)
  13. 13.
    Lewis, J.L.: Regularity of the derivatives of solutions to certain degenerate elliptic equations. Indiana Univ. Marh. J. 32, 849–858 (1983)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)zbMATHGoogle Scholar
  15. 15.
    Mtiri, F., Harrabi, A., Ye, D.: Explicit \(L^\infty \)-norm estimates via Morse index for the bi-harmonic and tri-harmonic semilinear problems. Manuscr. Math. (2018).
  16. 16.
    Ôtani, M.: Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations. J. Funct. Anal. 76, 140–159 (1988)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Quittner, P., Souplet, P.: A priori estimates and existence for elliptic systems via boot-strap in weighted Lebesgue spaces. Arch. Ration. Mech. Anal. 174, 49–81 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rahal, B., Harrabi, A.: Liouville results for \(m\)-laplace equations in half-space and strips with mixed boundary value conditions and finite Morse Index. J. Dyn. Differ. Equ.
  19. 19.
    Tolkswrf, P.: On the Dirichlet problem for quasilinear equations in domains with conical boundary points. Commun. Partial Differ. Equ. 8, 773–817 (1983)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yang, X.F.: Nodal sets and Morse Indices of solutions of super-linear elliptic PDE’s. Funct. Anal. 160, 223–253 (1998)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematics Department Faculty of Science of SfaxUniversity of SfaxSfaxTunisia
  2. 2.Military School of Aeronautical SpecialitiesSfaxTunisia
  3. 3.Mathematics DepartmentNorthern Borders UniversityArarSaudi Arabia
  4. 4.Institut Supérieur des MathématiquesAppliquées et de l’InformatiqueUniversité de KairouanTunisia
  5. 5.Abdus Salam International Centre for Theoretical PhysicsTriesteItaly

Personalised recommendations