Some issues on the p-Laplace equation in cylindrical domains

Abstract

In this article the authors prove a theorem regarding the convergence of solutions for the problems $$ \cases -\Delta_p u_l=f(X_2) &\text{ in $\Omega_l$},\\ u_l=0 &\text{ on $\partial \Omega_l$},\endcases $$ as $l\to \infty$. Here the $n$-dimensional space $\Bbb{R}^n$ is written as the product $\Bbb{R}^q\times \Bbb{R}^{n-q}$ and $X \in \Bbb{R}^n$ is written as $X = (X_1, X_2) = (x_1, \dots, x_q, x_{q+1}, \dots, x_n)$. The domain is $\Omega_l = (-l,l)^q \times \omega$ and $\omega$ is a smooth bounded domain in $\Bbb{R}^{n-q}$. The authors also present a Liouville-type nonexistence result on the domain ${\Omega_\infty = \Bbb{R}^q \times \omega}$.