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

We consider the inhomogeneous Allen-Cahn equation $${ϵ}^2\mathrm{\Delta }u+V(y)(1-u^2)u=0in\mathrm{\Omega },\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=0on\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^2$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ and $V(x)$ is a positive smooth function, $ϵ>0$ is a small parameter, $\nu$ denotes the unit outward normal of $\mathrm{\partial }\mathrm{\Omega }$. For any fixed integer $N\ge 2$, we will show the existence of a clustered solution $u_{ϵ}$ with $N$-transition layers near $\mathrm{\partial }\mathrm{\Omega }$ with mutual distance $O(ϵ\mathrm{\mid }\ln ϵ\mathrm{\mid })$, provided that the generalized mean curvature $\mathcal{H}$ of $\mathrm{\partial }\mathrm{\Omega }$ 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)