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Entire solutions in R2 for a class of Allen-Cahn equations by Alessio F., Montecchiari P.

By Alessio F., Montecchiari P.

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5 we deduce that there exists T > 0 such that if |y| > L then vp (x, y) ≥ 1 − 2δ for any x > T and vp (x, y) ≤ −1 + 2δ for any x < −T. 23) We will assume that T is such that also 1 − |z0 (x)| ≥ 2δ for any |x| ≥ T . Assume now by contradiction that vp (x, y) does not converge to ±1 as x → ±∞ uniformly with respect to y ∈ R. Then there exists a sequence (xj , yj ) ⊂ R2 such that |xj | → ∞, |yj | → ∞ as j → ∞ and 1 − |vp (xj , yj )| ≥ 2r > 0 for any j ∈ N. Since vp C 2 (R2 ) < +∞ we obtain that there exists ρ ∈ (0, 1) such that 1 − |vp (x, y)| ≥ r for any (x, y) ∈ ∪j∈N Bρ ((xj , yj )) (as usual we denote Bρ ((xj , yj )) = {(x, y) ∈ R2 / (x − xj )2 + (y − yj )2 < ρ2 }).

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