  This paper is devoted to the Lin-Ni conjecture for a semi-linear elliptic
equation with a super-linear, sub-critical nonlinearity and homogeneous Neumann
boundary conditions. We establish a new rigidity result, that is, we prove that
the unique positive solution is a constant if the parameter of the problem is
below an explicit bound that we relate with an optimal constant for a
Gagliardo-Nirenberg-Sobolev interpolation inequality and also with an optimal
Keller-Lieb-Thirring inequality. Our results are valid in a sub-linear regime
as well. The rigidity bound is obtained by nonlinear flow methods inspired by
recent results on compact manifolds, which unify nonlinear elliptic techniques
and the carr{\'e} du champ method in semi-group theory. Our method requires the
convexity of the domain. It relies on integral quantities, takes into account
spectral estimates and provides improved functional inequalities.
