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Zeitschrift für Analysis und ihre Anwendungen


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Volume 40, Issue 2, 2021, pp. 167–182
DOI: 10.4171/ZAA/1678

Published online: 2021-03-30

A Kirchhoff $p(x)$-Biharmonic Problem Involving Singular Nonlinearities and Navier Boundary Conditions

Khaled Kefi[1], Kamel Saoudi[2] and Mohammed Mosa AL-Shomrani[3]

(1) Université de Tunis El Manar, Tunisia
(2) Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia
(3) King Abdulaziz University, Jeddah, Saudi Arabia

The aim of this work is to study the existence of weak solutions for a nonhomogeneous singular $p(x)$-Kirchhoff problem of the following form \begin{equation*} (\textbf{P}_{\pm \lambda}) \quad \left\{ \begin{aligned} M(t)\Delta(|\Delta u|^{p(x)-2}\Delta u) &=a(x) u^{-\gamma (x)}\pm \lambda u^{q(x)-2}u, &\ &\mbox{in }\Omega, \\ \Delta u&=u=0, & \ &\mbox{on }\partial\Omega, \end{aligned} \right. \end{equation*} by using variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces.

Keywords: $p(x)$-Kirchhoff problem, singular problem, variational arguments, generalized Lebesgue Sobolev spaces

Kefi Khaled, Saoudi Kamel, AL-Shomrani Mohammed Mosa: A Kirchhoff $p(x)$-Biharmonic Problem Involving Singular Nonlinearities and Navier Boundary Conditions. Z. Anal. Anwend. 40 (2021), 167-182. doi: 10.4171/ZAA/1678