On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

Abstract

We study positive solutions of the following semilinear equation

epsilon 2 Delta((g) over bar)u - V(z)u + u(p) = o on M,

where (M, (g) over bar) is a compact smooth n-dimensional Riemannian manifold without boundary or the Euclidean space R-n, epsilon 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-k/n-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 integral(K) Vp+1/P-1-n-k/2 dvol, we prove the existence of a sequence epsilon = epsilon(j) -> 0 and positive solutions u(epsilon) 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 Schredinger equation and f -minimal submanifolds in manifolds with density.