Abstract
We consider a class of semilinear elliptic equations of the form
where \({a:\mathbb {R} \to \mathbb {R}}\) is a periodic, positive, even function and, in the simplest case, \({W : \mathbb {R} \to \mathbb {R}}\) is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the one dimensional heteroclinic problem
we show, via variational methods the existence of infinitely many geometrically distinct solutions u of (0.1) verifying u(x, y, z) → ± 1 as x → ± ∞ uniformly with respect to \({(y, z) \in \mathbb {R}^{2}}\) and such that \({\partial_{y}u \not \equiv0, \partial_{z}u \not\equiv 0}\) in \({\mathbb {R}^{3}}\) .
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Alessio, F., Montecchiari, P. Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in \({\mathbb {R}^{3}}\) . Calc. Var. 46, 591–622 (2013). https://doi.org/10.1007/s00526-012-0495-2
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DOI: https://doi.org/10.1007/s00526-012-0495-2