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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
Article
  • 141 Downloads

Abstract

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.

Keywords

Morse index elliptic estimates Pohozaev identity m-Laplacian operator 

Mathematics Subject Classification

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

Notes

Acknowledgements

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.

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Copyright information

© The Author(s) 2019

Open AccessThis article is 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, 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

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