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

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 $Ω$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p$ is greater than $1$, $\varepsilon>0$ is a small parameter, $V$ is a uniformly positive smooth potential on $\barΩ$, and $ν$ denotes the outward normal of $\partial Ω$. For two positive smooth functions ${\mathfrak a}_1(y), {\mathfrak a}_2(y)$ on $\barΩ$, the operator $\nabla_{{\mathfrak a}(y)}$ is given by $\nabla_{{\mathfrak a}(y)} u=\Bigg({\mathfrak a}_1(y)\frac{\partial u}{\partial y_1}, \, {\mathfrak a}_2(y)\frac{\partial u}{\partial y_2}\Bigg).$

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