  In this paper we propose a counterexample to the validity of the Comparison
Principle and of the Sub and Supersolution Method for nonlocal problems like
the stationary Kirchhoff Equation. This counterexample shows that in general
smooth bounded domains in any dimension, these properties cannot hold true if
the nonlinear nonlocal term $M(\|u\|^2)$ is somewhere increasing with respect
to the $H_0^1$-norm of the solution.
  Comparing with existing results, this fills a gap between known conditions on
$M$ that guarantee or prevent these properties, and leads to a condition which
is necessary and sufficient for the validity of the Comparison Principle.
