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Complex Analysis and Operator Theory

, Volume 13, Issue 1, pp 115–126 | Cite as

Existence of Positive Solution for Kirchhoff Problems

  • K. Ben Ali
  • M. Bezzarga
  • A. GhanmiEmail author
  • K. Kefi
Article
  • 97 Downloads

Abstract

In this work, we study the following Kirchhoff type problem
$$\begin{aligned} \begin{gathered} -\Big (a+b\int _{\Omega }|\nabla u|^pdx\Big )\Delta _p u =g(x)u^{-\gamma }+\lambda f(x,u),\quad \text {in }\Omega , \\ u=0, \quad \text {on }\partial \Omega , \end{gathered} \end{aligned}$$
where \(p\ge 2\), \(\Omega \) is a regular bounded domain in \(\mathbb {R}^N\), \((N\ge 3)\). Firstly, for \(p>2\), we prove under some appropriate conditions on the singularity and the nonlinearity the existence of nontrivial weak solution to this problem. For \(p=2\), we show, under supplementary condition, the positivity of this solution. Moreover, in the case \(\lambda =0\) we prove an uniqueness result. We use the variational method to prove our main results.

Keywords

Kirchhoff type equation Singularity problem Variational methods Resonance Positive solution Mountain pass lemma 

Mathematics Subject Classification

35B09 35B33 35J20 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • K. Ben Ali
    • 1
    • 3
  • M. Bezzarga
    • 4
  • A. Ghanmi
    • 2
    • 3
    Email author
  • K. Kefi
    • 3
    • 5
  1. 1.Jazan Technical CollegeJazanSaudi Arabia
  2. 2.Mathematics Department, Faculty of Sciences and Arts KhulaisUniversity of Jeddah, KSAJeddahSaudi Arabia
  3. 3.Mathematics DepartmentFaculty of Sciences Tunis El ManarTunisTunisia
  4. 4.Mathematics DepartmentIPEITTunisTunisia
  5. 5.Community College of RafhaNorthern border university, KSAArarSaudi Arabia

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