A two end family of solutions for the inhomogeneous Allen-Cahn equation in R2

In this work we construct a family of entire bounded solutions for the singularly perturbed inhomogeneous Allen-Cahn equation ε2div(a(x)∇u)−a(x)F′(u)=0 in R2, where ε→0. The nodal set of these solutions is close to a “nondegenerate” curve which is asymptotically two non-parallel straight lines. Here F′ is a double-well potential and a is a smooth positive function. We also provide example of curves and functions a where our result applies. This work is in connection with the results found in [5,7,10] handling the compact case.