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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https://boundaryvalueproblems.springeropen.com/articles/10.1186/s13661-023-01814-2
