Nodal solutions for Neumann systems with gradient dependence

Abstract

We consider the following convective Neumann systems:\begin{equation*}\left(\mathrm{S}\right)\qquad\left\{\begin{array}{ll}-Δ_{p_1}u_1+\frac{|\nabla u_1|^{p_1}}{u_1+δ_1}=f_1(x,u_1,u_2,\nabla u_1,\nabla u_2) & \text{in}\;Ω,\\ -Δ_{p_2}u_2+\frac{|\nabla u_2|^{p_2}}{u_2+δ_2}=f_2(x,u_1,u_2,\nabla u_1,\nabla u_2)&\text{in}\;Ω, \\ |\nabla u_1|^{p_1-2}\frac{\partial u_1}{\partial η}=0=|\nabla u_2|^{p_2-2}\frac{\partial u_2}{\partial η}&\text{on}\;\partialΩ,\end{array}\right.\end{equation*}where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ ($N\ge 2$) with a smooth boundary $\partial \Omega$,${\delta }_1,{\delta }_2>0$ are small parameters, $\eta$ is the outward unit vector normal to $\partial \Omega ,$ $f_1,f_2:\Omega \times {\mathbb{R}}^2\times {\mathbb{R}}^{2N}\to \mathbb{R}$ 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 every $u_i\in W^{1,p_i}(\Omega ).$ 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.

References in corpus (4)

• Nodal solutions to a Neumann problem for a class of (p_1,p_2)-Laplacian systems
• Strongly singular convective elliptic equations in RN driven by a non-homogeneous operator
• On a comparison theorem for parabolic equations with nonlinear boundary conditions
• Existence and location of nodal solutions for quasilinear convection-absorption Neumann problems