Existence and Nonexistence of Weak Positive Solution for a Class of p-Laplacian Systems

Abstract

In this work, we are interested to obtain some result of existence and nonexistence of positive weak solution for the following p-Laplacian system $$\begin{equation}\begin{case}-Δ_{pi}u_i=λ_if_i (u_1,…,u_m),\qquad in\;Ω,\;\;i=1,…,m,\\ui=0,\qquad on ∂Ω,\;\;∀i=1,…,m,\end{case}\end{equation}$$ where Δ_{pi}z=div(|∇z|^{pi-2}∇z), pi ≥ 1,λ_i,1 ≤ i ≤ m are a positive parameter, and Ω is a bounded domain in \mathbb{R}^N with smooth boundary ∂Ω. The proof of the main results is based to the method of sub-supersolutions.