We consider the inhomogeneous Allen–Cahn equation

where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary \(\partial \Omega \) and V(x) is a positive smooth function, \(\epsilon >0\) is a small parameter, \(\nu \) denotes the unit outward normal of \(\partial \Omega \). For any fixed integer \(N\ge 2\), we will show the existence of a clustered solution \(u_{\epsilon }\) with N-transition layers near \(\partial \Omega \) with mutual distance \(O(\epsilon |\ln \epsilon |)\), provided that the generalized mean curvature \(\mathcal {H} \) of \(\partial \Omega \) is positive and \(\epsilon \) stays away from a discrete set of values at which resonance occurs. Our result is an extension of those (with dimension two) by Malchiodi et al. (Pac. J. Math. 229(2):447–468, 2007) and Malchiodi et al. (J. Fixed Point Theory Appl. 1(2):305–336, 2007).

我们考虑了非均质的艾伦-卡恩方程

其中\（\ Omega \）是\（{\ mathbb {R}} ^ 2 \）中具有光滑边界\（\ partial \ Omega \）的有界域，而V（x）是正光滑函数，\（\ epsilon> 0 \）是一个小参数，\（\ nu \）表示\（\ partial \ Omega \）的单位向外法线。对于任何固定整数\（N \ ge 2 \），我们将显示存在聚类解\（u _ {\ epsilon} \）的存在，其中N个过渡层靠近\（\ partial \ Omega \），并且相互距离\（ O（\ epsilon | \ ln \ epsilon |）\），条件是广义平均曲率\（\ mathcal {H} \）的\（\局部\欧米茄\）是正的和\（\小量\）从一组离散值中的哪个发生共振停留程。我们的结果是对Malchiodi等人（具有第二个维度）的扩展。（Pac。J. Math。229（2）：447-468，2007）和Malchiodi等。（J.不动点理论研究，Appl。1（2）：305-336，2007）。