Alessio, Francesca; Jeanjean, Louis; Montecchiari, Piero Existence of infinitely many stationary layered solutions in \(\mathbb R^2\) for a class of periodic Allen-Cahn equations. (English) Zbl 1125.35342 Commun. Partial Differ. Equations 27, No. 7-8, 1537-1574 (2002). Variational methods are used to find solutions of the semilinear elliptic equation \[ -\Delta u + a(x,y) W'(u)=0, \quad (x,y)\in\mathbb{R}^2 \] where \(a\) is periodic in both variables and \(W\) is a double-well potential. The main result states that there are infinitely many different (up to periodic translations in \(x\) and \(y\)) solutions. This is proved in three steps. First, the existence of solutions which are periodic in \(y\) and which decay to constant states as \(x\to\pm\infty\) is established. If there are only finitely many of these periodic solutions (up to translations in \(x\)) then in a second step the existence of heteroclinic orbits is shown which connect different periodic profiles as \(y\to\pm\infty\). In a last step, it is shown that if the number of these heteroclinic orbits is finite (up to translations in \(y\)) then there exist infinitely many multibump solutions. Reviewer: Jörg Härterich (Berlin) Cited in 23 Documents MSC: 35J60 Nonlinear elliptic equations 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 35B10 Periodic solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 35J20 Variational methods for second-order elliptic equations 47J30 Variational methods involving nonlinear operators Keywords:variational methods; multibump solutions; heteroclinic solutions; concentration compactness method PDFBibTeX XMLCite \textit{F. Alessio} et al., Commun. Partial Differ. Equations 27, No. 7--8, 1537--1574 (2002; Zbl 1125.35342) Full Text: DOI References: [1] DOI: 10.1007/s005260050071 · Zbl 0883.35036 · doi:10.1007/s005260050071 [2] Alberti G., De Giorgi: Old and Recent Results [3] Alessio F., Z. Angew. Math. Phys. 50 (6) pp 860– (1999) · Zbl 0960.70018 · doi:10.1007/s000330050184 [4] Alessio F., Progr. Nonlinear Differential Equations Appl., 43, in: Nonlinear Analysis and its Applications to Differential Equations (Lisbon, 1998), 147–159 (2001) [5] DOI: 10.1007/s005260000036 · Zbl 0965.35050 · doi:10.1007/s005260000036 [6] DOI: 10.1016/0001-6160(79)90196-2 · doi:10.1016/0001-6160(79)90196-2 [7] Ambrosio L., J. Am. Math. Soc. 13 (4) pp 725– (2000) · Zbl 0968.35041 · doi:10.1090/S0894-0347-00-00345-3 [8] Barlow M.T., Comm. Pure Appl. Math. 53 (8) pp 1007– (2000) · Zbl 1072.35526 · doi:10.1002/1097-0312(200008)53:8<1007::AID-CPA3>3.0.CO;2-U [9] Berestycki H., Duke Math. J. 103 (3) pp 375– (2000) · Zbl 0954.35056 · doi:10.1215/S0012-7094-00-10331-6 [10] De Giorgi E., Proc. Int. Meeting on Recent Methods in Nonlinear Analysis (1978) [11] Farina A., Ricerche di Matematica (in memory of Ennio De Giorgi) 48 pp 129– (1999) [12] DOI: 10.1007/s002080050196 · Zbl 0918.35046 · doi:10.1007/s002080050196 [13] Nirenberg L., CBMS Reg. Conf. Ser. in Math., AMS 17 (1973) · doi:10.1090/cbms/017 [14] Rabinowitz P.H., Ergod. Th. and Dyn. Sys. 14 pp 817– (1994) [15] Rabinowitz P.H., J. Math. Sci. Univ. Tokio 1 pp 525– (1994) [16] Rabinowitz P.H., Ergodic Theory Dynam. Systems 20 (6) pp 1767– (2000) · Zbl 0981.37020 · doi:10.1017/S0143385700000985 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.