On the Ambrosetti-Malchiodi-Ni Conjecture for general submanifolds

We study positive solutions of the following semilinear equation $${\epsilon }^2{\mathrm{\Delta }}_{\bar{g}}u-V(z)u+u^p=0\text{ on }M,$$ where $(M,\bar{g})$ is a compact smooth $n$-dimensional Riemannian manifold without boundary or the Euclidean space ${\mathbb{R}}^n$, $\epsilon$ is a small positive parameter, $p>1$ and $V$ is a uniformly positive smooth potential. Given $k=1,\dots ,n-1$, and $1<p<\frac{n+2-k}{n-2-k}$. Assuming that $K$ is a $k$-dimensional smooth, embedded compact submanifold of $M$, which is stationary and non-degenerate with respect to the functional $\int_K V^{\frac{p+1}{p-1}-\frac{n-k}{2}}dvol$,weprovetheexistenceofasequence$\varepsilon=\varepsilon_j\to 0$andpositivesolutions$u_\varepsilon$ that concentrate along $K$. This result proves in particular the validity of a conjecture by Ambrosetti-Malchiodi-Ni, extending a recent result by Wang-Wei-Yang, where the one co-dimensional case has been considered. Furthermore, our approach explores a connection between solutions of the nonlinear Schr"{o}dinger equation and $f$-minimal submanifolds in manifolds with density.