Nodal solutions for Neumann systems with gradient dependence

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

We consider the following convective Neumann systems: $\begin{equation*}\left(\mathrm{S}\right)\qquad\left\{\begin{array}{ll}-\Delta_{p_1}u_1+\frac{|\nabla u_1|^{p_1}}{u_1+\delta_1}=f_1(x,u_1,u_2,\nabla u_1,\nabla u_2) \text{in}\;\Omega,\\ -\Delta _{p_2}u_2+\frac{|\nabla u_2|^{p_2}}{u_2+\delta_2}=f_2(x,u_1,u_2,\nabla u_1,\nabla u_2) \text{in}\;\Omega, \\ |\nabla u_1|^{p_1-2}\frac{\partial u_1}{\partial \eta }=0=|\nabla u_2|^{p_2-2}\frac{\partial u_2}{\partial \eta} \text{on}\;\partial\,\Omega,\end{array}\right.\end{equation*}$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$) with a smooth boundary $\partial\,\Omega, \delta_1, \delta_2 > 0$ are small parameters, $\eta$ is the outward unit vector normal to $\partial\,\Omega, f_1, f_2: \Omega \times \mathbb{R}^2 \times \mathbb{R}^{2N} \rightarrow \mathbb{R}$ are Carathéodory functions that satisfy certain growth conditions, and $\Delta _{p_i}$ ($1< p_i < N,$ for $i=1,2$) are the $p$-Laplace operators $\Delta _{p_i}u_i=\mathrm{div}(|\nabla u_i|^{p_i-2}\nabla u_i)$, for $u_i \in W^{1,p_i}(\Omega).$ In order to prove the existence of solutions to such systems, we use a sub-supersolution method. We also obtain nodal solutions by constructing appropriate sub-solution and super-solution pairs. To the best of our knowledge, such systems have not been studied yet.

• Language: English
• Keywords: Neumann elliptic systems, gradient dependence, subsolution method, supersolution method, nodal 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: 2024
• Number of pages: 19 str.
• Numbering: Vol. 2024, article no. 4
• PID: 20.500.12556/RUL-153500
• UDC: 517.9
• ISSN on article: 1687-2770
• DOI: 10.1186/s13661-023-01814-2
• COBISS.SI-ID: 180215555
• Publication date in RUL: 10.01.2024

Record is a part of a journal

• Title: Boundary value problems
• Shortened title: Bound. value probl.
• Publisher: Springer
• ISSN: 1687-2770
• COBISS.SI-ID: 62113025

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• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
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