Rigidity for a class of partial differential equations

This paper investigates rigidity properties of certain elliptic PDEs on smooth compact connected Riemannian manifolds without boundaries, along with optimal constants in associated Sobolev-type interpolation inequalities. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. The maximal parameter value yielding this rigidity provides an estimate for the sharp constant in the corresponding interpolation inequality.