Nodal solutions for Neumann systems with gradient dependence

Abstract

Abstract We consider the following convective Neumann systems: ( S ) { − Δ p 1 u 1 + | ∇ u 1 | p 1 u 1 + δ 1 = f 1 ( x , u 1 , u 2 , ∇ u 1 , ∇ u 2 ) in Ω , − Δ p 2 u 2 + | ∇ u 2 | p 2 u 2 + δ 2 = f 2 ( x , u 1 , u 2 , ∇ u 1 , ∇ u 2 ) in Ω , | ∇ u 1 | p 1 − 2 ∂ u 1 ∂ η = 0 = | ∇ u 2 | p 2 − 2 ∂ u 2 ∂ η on ∂ Ω , $$ ( \mathrm{S} ) \quad \textstyle\begin{cases} -\Delta _{p_{1}}u_{1}+ \frac{ \vert \nabla u_{1} \vert ^{p_{1}}}{u_{1}+\delta _{1} }=f_{1}(x,u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) & \text{in } \Omega , \\ -\Delta _{p_{2}}u_{2}+ \frac{ \vert \nabla u_{2} \vert ^{p_{2}}}{u_{2}+\delta _{2} }=f_{2}(x,u_{1},u_{2}, \nabla u_{1},\nabla u_{2}) & \text{in } \Omega , \\ \vert \nabla u_{1} \vert ^{p_{1}-2}\frac{\partial u_{1}}{\partial \eta }=0= \vert \nabla u_{2} \vert ^{p_{2}-2}\frac{\partial u_{2}}{\partial \eta } & \text{on } \partial \Omega ,\end{cases} $$ where Ω is a bounded domain in R N $\mathbb{R}^{N}$ ( N ≥ 2 $N\geq 2$ ) with a smooth boundary ∂Ω, δ 1 , δ 2 > 0 $\delta _{1}, \delta _{2} >0$ are small parameters, η is the outward unit vector normal to ∂Ω, f 1 , f 2 : Ω × R 2 × R 2 N → R $f_{1}, f_{2}:\Omega \times \mathbb{R}^{2}\times \mathbb{R}^{2N} \rightarrow \mathbb{R}$ are Carathéodory functions that satisfy certain growth conditions, and Δ p i $\Delta _{p_{i}}$ ( 1 < p i < N $1< p_{i}< N$ , i = 1 , 2 $i=1,2$ ) are the p-Laplace operators Δ p i u i = div ( | ∇ u i | p i − 2 ∇ u i ) $\Delta _{p_{i}}u_{i}=\operatorname{div}(|\nabla u_{i}|^{p_{i}-2}\nabla u_{i})$ for u i ∈ W 1 , p i ( Ω ) $u_{i}\in W^{1,p_{i}}(\Omega )$ . To prove the existence of solutions to such systems, we use a subsupersolution method. We also obtain nodal solutions by constructing appropriate subsolution and supersolution pairs. To the best of our knowledge, such systems have not been studied yet.

Keywords

• Neumann elliptic system
• Gradient dependence
• Subsolution and supersolution method
• Nodal solution