Clustering of Boundary Interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain

We consider the inhomogeneous Allen-Cahn equation $$ ε^2Δu\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ Ω, \qquad \frac {\partial u}{\partial ν}\,=\,0\quad \mbox{on}\ \partial Ω, $$ where $Ω$ is a bounded domain in ${\mathbb R}^2$ with smooth boundary $\partialΩ$ and $V(x)$ is a positive smooth function, $ε>0$ is a small parameter, $ν$ denotes the unit outward normal of $\partialΩ$. For any fixed integer $N\geq 2$, we will show the existence of a clustered solution $u_ε$ with $N$-transition layers near $\partial Ω$ with mutual distance $O(ε|\ln ε|)$, provided that the generalized mean curvature $\mathcal{H}$ of $\partialΩ$ is positive and $ε$ stays away from a discrete set of values at which resonance occurs. Our result is an extension of those (with dimension two) by A. Malchiodi, W.-M. Ni, J. Wei in Pacific J. Math. (Vol. 229, 2007, no. 2, 447-468) and A. Malchiodi, J. Wei in J. Fixed Point Theory Appl. (Vol. 1, 2007, no. 2, 305-336)

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