On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

We study positive solutions of the following semilinear equationε2Δg¯u−V(z)u+up=0on M, where (M,g¯) is a compact smooth n  -dimensional Riemannian manifold without boundary or the Euclidean space RnRn, ε   is a small positive parameter, p>1p>1 and V   is a uniformly positive smooth potential. Given k=1,…,n−1k=1,…,n−1, and 1