In this paper, we study a system of the form −normalΔλu=v−normalΔλv=upinℝN, where p∈ℝ, and Δλ is a sub‐elliptic operator defined by Δλ=∑i=1N∂xiλi2∂xi. Under some general hypotheses of the functions λi, i=1,2,…,N, we first prove that the system has no positive super‐solution when p ≤ 1. In the case p > 1, we establish a Liouville type theorem for the class of stable positive solutions. This result is an extension of some result in Hajlaoui et al. (Pacific J Math. 2014;270(1):79–93) for the case of Laplace operator.

We first show that the minimization problem associated with the nonlinear shell model of W.T. Koiter becomes coercive over its natural space of admissible deformations when the third fundamental form is added to its functional. Then, under the assumption that the middle surface of the shell is a minimal surface, we approach this minimization problem by a new minimization problem that is well-posed over the same space of admissible deformations.