On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: clustering concentration layers

Abstract

We consider the clustering concentration on curves for solutions to the problem $$ \varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0 \quad\mbox{in }Ω, \qquad \nabla_{{\mathfrak a}(y)} u\cdot ν\, =\, 0\quad\mbox{on } \partial Ω, $$ where $\Omega$ is a bounded domain in ${\mathbb{R}}^2$ with smooth boundary, the exponent $p$ is greater than $1$, $\epsilon >0$ is a small parameter, $V$ is a uniformly positive smooth potential on $\bar{\Omega }$, and $\nu$ denotes the outward normal of $\partial \Omega$. For two positive smooth functions ${\mathfrak{a}}_1(y),{\mathfrak{a}}_2(y)$ on $\bar{\Omega }$, the operator ${\nabla }_{\mathfrak{a}(y)}$ is given by $${\nabla }_{\mathfrak{a}(y)}u=({\mathfrak{a}}_1(y)\frac{\partial u}{\partial y_1},{\mathfrak{a}}_2(y)\frac{\partial u}{\partial y_2}).$$

arXiv admin note: text overlap with arXiv:1603.07175