Summary: We consider the problem \(\varepsilon^2 \Delta u+ (u-a(x))(1-u^2)=0\) in \(\Omega\), \(\frac{\partial u}{\partial \nu}=0\) on \(\partial \Omega\), where \(\Omega\) is a smooth and bounded domain in \(\mathbb R^2\), \(-1<a(x) <1\). Assume that \(\Gamma = \{ x \in \Omega\), \(a(x)=0 \}\) is a closed, smooth curve contained in \(\Omega\) in such a way that \(\Omega = \Omega_+\cup\Gamma\cup \Omega_-\) and \(\frac{\partial a}{\partial n}>0\) on \(\Gamma\), where \(n\) is the outer normal to \(\Omega_+\). P.C. Fife and W.M. Greenlee [Russ. Math. Surv. 29, No. 4, 103–131 (1974); translation from Usp. Mat. Nauk 29, No. 4(178), 103–130 (1974; Zbl 0309.35035)] proved the existence of an interior transition layer solution \(u_\varepsilon\) which approaches \(-1\) in \(\Omega_-\) and \(+1\) in \(\Omega_+\), for all \(\varepsilon\) sufficiently small. A question open for many years has been whether an interior transition layer solution approaching 1 in \(\Omega_-\) and \(-1\) in \(\Omega_+\) exists.
In this paper, we answer this question affirmatively when \(n=2\), provided that \(\varepsilon\) is small and away from certain critical numbers. A main difficulty is a resonance phenomenon induced by a large number of small critical eigenvalues of the linearized operator.