Existence and multiplicity results for a new ▫$p(x)$▫-Kirchhoff problem

Hamdani, Mohamed Karim (author), Harrabi, Abdellaziz (author), Mtiri, Foued (author), Repovš, Dušan (author)

In this work, we study the existence and multiplicity results for the following nonlocal-Kirchhoff problem: ▫$$\begin{cases} -\big(a-b \int_\Omega \frac{1}{p(x}|\nabla u|^{p(x)} dx \big) \; \text{div} (|\nabla u|^{p(x)-2} \nabla u) = \\ = \lambda |u|^{p(x)-2}u + g(x,u) & \text{in} \; \Omega \\ u=0 & \text{on} \; \partial \Omega \end{cases}$$▫ where ▫$a \ge b > 0$▫ are constants, ▫$\Omega \subset \mathbb{R}^N$▫ is a bounded smooth domain ▫$p \in C(\overline{\Omega})$▫, with ▫$N > p(x) > 1$▫, ▫$\lambda$▫ is a real parameter and ▫$g$▫ is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.

• Language: English
• Keywords: variable exponent, nonlocal Kirchhoff equation, p(x)-Laplacian operator, Palais-Smale condition, Mountain Pass theorem, Fountain theorem
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2020
• Number of pages: art. 111598 ( 15 str.)
• Numbering: Vol. 190
• PID: 20.500.12556/RUL-109278
• UDC: 517.956
• ISSN on article: 0362-546X
• DOI: 10.1016/j.na.2019.111598
• COBISS.SI-ID: 18706265
• Publication date in RUL: 29.08.2019

Record is a part of a journal

• Title: Nonlinear Analysis
• Shortened title: Nonlinear anal.
• Publisher: Pergamon Press
• ISSN: 0362-546X
• COBISS.SI-ID: 26027520