On the Ambrosetti-Malchiodi-Ni Conjecture for general submanifolds

Abstract

We study positive solutions of the following semilinear equation $$\varepsilon^2\Delta_{\bar g} u - V(z) u+ u^{p} =0\,\hbox{ on }\,M, $$ where $(M, \bar g )$ is a compact smooth $n$-dimensional Riemannian manifold without boundary or the Euclidean space $\mathbb R^n$, $\varepsilon$ 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 $\int_K V^{\frac{p+1}{p-1}-\frac{n-k}{2}}dvol$, we prove the existence of a sequence $\varepsilon=\varepsilon_j\to 0$ and positive solutions $u_\varepsilon$ 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.