Uniqueness of bounded solutions to p-Laplace problems in strips

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 ∈(0,+∞], 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 <+∞ 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]