Let W be a non-negative function of class C3 from $\xR^2$ to $\xR$ , which vanishes exactly at two points a and b. LetS 1(a, b) be the set of functions of a real variable which tend to a at -∞and to b at +∞ and whose one dimensional energy $$E_1(v)=\int_\xR\bigl[W(v)+\lvert v'\rvert^2/2\bigr]\,\xdif x$$ is finite.Assume that there exist two isolated minimizers z + and z - of the energy E 1over S 1(a, b). Under a mild coercivity condition on thepotential W and a generic spectral condition on the linearization of theone-dimensional Euler–Lagrange operator at z + and z - , it ispossible to prove that there exists a function ufrom $\xR^2$ to itself which satisfies the equation $$-\Delta u + \xDif W(u)^\mathsf{T}=0,$$ and the boundary conditions $$\lim_{x_2\to +\infty} u(x_1,x_2)=z_+(x_1-m_+),\phantom{\mathbf{a}}\lim_{x_2\to-\infty} u(x_1,x_2)=z_-(x_1-m_-),\lim_{x_1\to -\infty}u(x_1,x_2)=\mathbf{a},\phantom{z_+(x_1-m_+)} \lim_{x_1\to+\infty}u(x_1,x_2)=\mathbf{b}.$$ The above convergences are exponentially fast; the numbers m + and m - are unknowns of the problem.