Ambrosetti–Malchiodi–Ni concentration-layer conjecture

Let $\Gamma$ be a $k$-dimensional submanifold in $\mathbb R^\mathcal{N}$ and a nondegenerate critical point of the functional

$$\mathcal{K}(\Gamma)=\int_\Gamma V^{\frac{p+1}{p-1}-\frac12(\mathcal{N}-k)}\,dvol.$$

Assume $1<p<\frac{n+2-k}{n-2+k}$ for $\mathcal{N}\ge 3$ and $p>1$ for $\mathcal{N}=2$. Ambrosetti–Malchiodi–Ni conjecture. There exists a family of solutions to

$$-\varepsilon^2\Delta u+V(y)u=u^p \qquad\text{in }\mathbb R^\mathcal{N}$$

concentrating along $\Gamma$ at least for a subsequence $\varepsilon=\varepsilon_j\to0$. This conjecture predicts concentration of solutions to the nonlinear Schrödinger equation along nondegenerate critical submanifolds; the source presents it as a conjecture arising from the analogy with high-dimensional concentration phenomena for the Ambrosetti–Prodi type problem.