It is proved that the Cahn-Hilliard equation on a smooth domain possesses solutions which have spike layers localizing where the mean curvature of the boundary of the domain has nondegenerate critical points. Solutions of this type can be found with any average value which lies in the metastable region. It is also shown that these solutions have Morse indices at least equal to the number of spikes.

Elliptic systems with noncooperative coupling in general do not preserve positivity. Here it is shown that some systems of elliptic equations on $ \mathbb{R}^N$ with small noncooperative coupling still have a restricted uniform positivity preserving property. The proofs rely on optimal estimates for the Newtonian potential with weights and on corresponding $3G$-type theorems.