On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

Abstract

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 Rn, ε is a small positive parameter, p>1 and V is a uniformly positive smooth potential. Given k=1,…,n−1, and 1<p<n+2−kn−2−k. Assuming that K is a k-dimensional smooth, embedded compact submanifold of M, which is stationary and non-degenerate with respect to the functional ∫KVp+1p−1−n−k2dvol, we prove the existence of a sequence ε=εj→0 and positive solutions uε that concentrate along K. This result proves in particular the validity of a conjecture by Ambrosetti et al. [1], extending a recent result by Wang et al. [32], where the one co-dimensional case has been considered. Furthermore, our approach explores a connection between solutions of the nonlinear Schrödinger equation and f-minimal submanifolds in manifolds with density.