The Nehari manifold approach for singular equations involving the $p(x)$-Laplace operator

Repovš, Dušan (author), Saoudi, Kamel (author)

We study the following singular problem involving the $p(x)$-Laplace operator $\Delta p(x)u = \mathrm{div}(|\nabla u|^{p(x)-2} \nabla u)$, where $p(x)$ is a nonconstant continuous function $$ (P_\lambda) \quad \begin{cases} -\Delta_{p(x)}u = a(x)|u|^{q(x)-2}u(x) + \frac{\lambda b(x)}{u^{\delta(x)}} & \text{in} \; \Omega, \\ u>0 & \text{in} \; \Omega, \\ u=0 &\text{on} \; \partial\Omega. \end{cases} $$ Here, $\Omega$ is a bounded domain in $\mathbb{R}^{N \ge 2}$ with $C^2$-boundary, $\lambda$ is a positive parameter, $a(x), b(x) \in C(\overline{\Omega})$ are positive weight functions with compact support in $\Omega$, and $\delta(x), p(x), q(x) \in C(\overline{\Omega})$ satisfy certain hypotheses $(A_0)$ and $(A_1)$. We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem $(P_\lambda)$.

• Language: English
• Keywords: Nehari manifold, generalized Lebesgue-Sobolev space, topological method, singular equation, p(x)-Laplace operator, multiplicity
• 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: Author Accepted Manuscript
• Publication date: 01.01.2023
• Year: 2023
• Number of pages: Str. 135-149
• Numbering: Vol. 68, no. 1
• PID: 20.500.12556/RUL-182455
• UDC: 517.956
• ISSN on article: 1747-6933
• DOI: 10.1080/17476933.2021.1980878
• COBISS.SI-ID: 80237571
• Publication date in RUL: 12.05.2026

Record is a part of a journal

• Title: Complex variables and elliptic equations
• Publisher: Taylor & Francis
• ISSN: 1747-6933
• COBISS.SI-ID: 513019929

Licences

• License: CC BY-NC 4.0, Creative Commons Attribution-NonCommercial 4.0 International
• Link: http://creativecommons.org/licenses/by-nc/4.0/
• Description: A creative commons license that bans commercial use, but the users don’t have to license their derivative works on the same terms.