Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains. (English) Zbl 1273.35132

Ackermann, Nils; Clapp, Mónica; Pacella, Filomena

Commun. Partial Differ. Equations 38, No. 4-6, 751-779 (2013).

Summary: Let \(\Gamma\) denote a smooth simple curve in \(\mathbb R^N, N\geq 2\), possibly with boundary. Let \(\Omega_R\) be the open normal tubular neighborhood of radius 1 of the expanded curve \(R\Gamma :=\{Rx|x\in\Gamma\setminus\Gamma\}\). Consider the superlinear problem \(-\Delta u+\lambda u=f(u)\) on the domains \(\Omega_R\), as \(R\to\infty\), with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along \(R\Gamma\) with alternating signs. The function \(f\) is superlinear at 0 and at \(\infty\), but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.

MSC:

35J61	Semilinear elliptic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

expanding domain; multibump solutions; semilinear elliptic equation; sign changing solutions

References:

[1]	DOI: 10.1016/j.jfa.2005.11.010 · Zbl 1126.35057 · doi:10.1016/j.jfa.2005.11.010
[2]	DOI: 10.1007/s00032-011-0147-6 · Zbl 1229.35285 · doi:10.1007/s00032-011-0147-6
[3]	Bartsch T., Topol. Methods Nonlinear Anal. 22 pp 1– (2003) · Zbl 1094.35041 · doi:10.12775/TMNA.2003.025
[4]	DOI: 10.1215/S0012-7094-96-08117-X · Zbl 0860.35004 · doi:10.1215/S0012-7094-96-08117-X
[5]	Berestycki H., Some Qualitative Properties of Solutions of Semilinear Elliptic Equations in Cylindrical Domains (1990) · Zbl 0705.35004 · doi:10.1016/B978-0-12-574249-8.50011-0
[6]	DOI: 10.1216/rmjm/1181071858 · Zbl 0907.35050 · doi:10.1216/rmjm/1181071858
[7]	DOI: 10.1016/0022-0396(88)90021-6 · Zbl 0662.34025 · doi:10.1016/0022-0396(88)90021-6
[8]	DOI: 10.1016/0022-0396(90)90005-A · Zbl 0729.35050 · doi:10.1016/0022-0396(90)90005-A
[9]	Dancer E.N., Math. Ann. 325 pp 369– (203) · Zbl 1040.35033 · doi:10.1007/s00208-002-0352-2
[10]	DOI: 10.1081/PDE-120002782 · Zbl 1011.35059 · doi:10.1081/PDE-120002782
[11]	DOI: 10.1090/S0002-9939-1993-1156467-3 · doi:10.1090/S0002-9939-1993-1156467-3
[12]	Gilbarg D., Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften 224, 2. ed. (1983) · Zbl 0562.35001 · doi:10.1007/978-3-642-61798-0
[13]	Spain P.G., Math. Mag. 69 pp 131– (1996)
[14]	DOI: 10.1007/BF01230289 · Zbl 0646.35034 · doi:10.1007/BF01230289
[15]	DOI: 10.1007/978-1-4612-1015-3 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3