| Title: | Nodal solutions for Neumann systems with gradient dependence |
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| Authors: | ID Saoudi, Kamel (Author)
ID Alzahrani, Eadah (Author)
ID Repovš, Dušan (Author) |
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| Files: |  URL - Source URL, visit https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-023-01814-2
 PDF - Presentation file, download (1,48 MB)
MD5: 0B6F5CF615DF90828E822E1CBBE3355D
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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. |
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| Keywords: | Neumann elliptic systems, gradient dependence, subsolution method, supersolution method, nodal solutions |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | 19 str. |
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| Numbering: | Vol. 2024, article no. 4 |
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| PID: | 20.500.12556/DiRROS-18198  |
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| UDC: | 517.9 |
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| ISSN on article: | 1687-2770 |
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| DOI: | 10.1186/s13661-023-01814-2 |
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| COBISS.SI-ID: | 180215555 |
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| Note: |
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| Publication date in DiRROS: | 16.02.2024 |
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| Views: | 640 |
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| Downloads: | 227 |
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