Heteroclinic solutions of Allen-Cahn type equations with a general elliptic operator (Q1741798)

In this paper, the following general type of the Allen-Cahn equation is considered: \[ -\operatorname{div}(\nabla G(\nabla u(x,y)))+F_u(x,y,u(x,y))=0\tag{AC} \] where \(u: \mathbb{R}^2 \rightarrow \mathbb{R}\) and \(F\in C^2(\mathbb{R}^3;\mathbb{R})\) satisfies some customary conditions for an Allen-Cahn problem, \(G\in C^{2,\alpha}(\mathbb{R}^2,\mathbb{R})\) satisfies certain standard growth and ellipticity conditions. Motivated by the paper [\textit{P. H. Rabinowitz} and \textit{E. W. Stredulinsky}, Commun. Pure Appl. Math. 56, No. 8, 1078--1134 (2003; Zbl 1274.35122)], the authors prove that the equation (AC) has a classical solution \(v(x,y)\in (0,1)\), which is is heteroclinic in \(x\) and periodic in \(y\), by using variational methods.