On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory

  • S. Taarabti Nat. School Appl. Sci. Agadir Ibn Zohr Univ., Morocco https://orcid.org/0000-0002-3134-9091
  • Z. El Allali Multidisciplinary Faculty of Nador, Mohammed First Univ., Oujda, Morocco
  • K. Ben Haddouch Nat. School Appl. Sci. Fes Sidi Mohammed Ben Abdellah Univ., Morocco

Abstract

UDC 517.9

The paper deals with the existence and multiplicity of nontrivial weak solutions for the $p(x)$-Kirchhoff-type problem

$$ {-M}\!\left(\displaystyle\int\limits_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\!\Delta_{p(x)}^{2} u = f(x,u)\quad \mbox{in}\quad \Omega, $$

$$ u = \Delta u = 0\quad  \mbox{on}\quad \partial\Omega.$$

By using variational approach and Krasnoselskii's genus theory, we prove the existence and multiplicity of solutions for the $p(x)$-Kirchhoff-type equation. 

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Published
17.06.2020
How to Cite
TaarabtiS., El Allali Z., and Ben HaddouchK. “On $\mathcal{p}(x)$-Kirchhoff-Type Equation Involving $\mathcal{p}(x)$-Biharmonic Operator via Genus Theory”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 6, June 2020, pp. 842-51, doi:10.37863/umzh.v72i6.6019.
Section
Research articles