The Lin–Ni conjecture in negative geometries

We prove (see Theorem 1.1) that the Lin–Ni conjecture for closed manifolds is false in dimensions n=4,5n=4,5 when the scalar curvature is negative, but that it holds true in the bounded energy setting when n≥6n≥6 (see Theorem 1.2). As a corollary to Theorem 1.1, we prove (see Theorem 2.2) that, contrary to the 3-dimensional case, the sup⁡×infsup⁡×inf inequality does not hold in general, with a sole control on the size of the potential, for positive solutions of stationary critical Schrödinger equations.