A TWO END FAMILY OF SOLUTIONS FOR THE INHOMOGENEOUS ALLEN-CAHN EQUATION IN R2

Abstract. In this work we construct a family of entire bounded solution for the singulary perturbed Inhomogeneous Allen-Cahn Equation ε2 div (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 paralell 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 [9], [7] and [12] handling the compact case. 1