New class of sixth-order nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions

Hamdani, Mohamed Karim (author), Chung, Nguyen Thanh (author), Repovš, Dušan (author)

In this paper, we prove the existence of multiple solutions for the following sixth-order $p(x)$-Kirchhoff-type problem $$\begin{cases} -M\left( \int\limits_{\it \Omega} \frac{1}{p(x)}|\nabla {\it\Delta} u|^{p(x)}dx\right){\it\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) &\mbox{in}\quad {\it\Omega}, \\ u = {\it\Delta} u = {\it\Delta}^2 u = 0, \quad &\mbox{on}\quad \partial{\it\Omega}, \end{cases}$$ where ${\it\Omega} \subset \mathbb{R}^N$ is a smooth bounded domain, $N>3$, ${\it\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div} \Big({\it\Delta}(|\nabla {\it\Delta} u|^{p(x)-2}\nabla {\it\Delta} u)\Big)$ is the $p(x)$-triharmonic operator, $p, q, r \in C(\overline{\it\Omega}), 1 < p ( x ) < \frac{N}{3}$ for all $x \in \overline{\it \Omega}, M(s) = a-bs^\gamma, \;a, b, \gamma > 0, \lambda > 0$, $g \colon {\it\Omega} \times \mathbb{R} \to \mathbb{R}$ is a nonnegative continuous function while $f, h \colon {\it\Omega} \times \mathbb{R} \to \mathbb{R}$ are sign-changing continuous functions in ${\it \Omega}$. To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order $p(x)$-Kirchhoff type problems with sign changing Kirchhoff functions.

• Language: English
• Keywords: variable exponents, Kirchhoff type problems, p(x)-triharmonic operator, sign-changing functions, concave-convex terms, Ekeland's variational principle, multiple solutions
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2021
• Number of pages: Str. 1117-1131
• Numbering: Vol. 10, iss. 1
• PID: 20.500.12556/RUL-128541
• UDC: 517.956
• ISSN on article: 2191-9496
• DOI: 10.1515/anona-2020-0172
• COBISS.SI-ID: 58245891
• Publication date in RUL: 19.07.2021

Record is a part of a journal

• Title: Advances in nonlinear analysis
• Publisher: De Gruyter
• ISSN: 2191-9496
• COBISS.SI-ID: 16253785

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• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.