Uniqueness of bounded solutions to $p$-Laplace problems in strips

Abstract

We consider a $p$-Laplace problem in a strip with two-constant boundary Dirichlet conditions. We show that if the width of the strip is smaller than some $d_0\in (0,+\infty ]$, then the problem admits a unique bounded solution, which is strictly monotone. Hence this unique solution is one-dimensional symmetric and belongs to the $C^2$ class. We also show that the problem has no bounded solution in the case that $d_0<+\infty $ and the width of the strip is larger than or equal to $d_0$. An analogous rigidity result in the whole space was obtained recently by Esposito et al. [8]

Keywords

• $p$-Laplace equation
• uniqueness
• monotonicity
• 1D symmetry