Oscillations in a semi-linear singular Sturm–Liouville equation

Article type: Research Article

Authors: Castro, Hernán

Affiliations: Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile. E-mails: hcastro@inst-mat.utalca.cl, hcastro.cl@gmail.com

Abstract: We study the semi-linear Sturm–Liouville equation −(x2αu′)′=λu+|u|p−1uin (0,1),u(1)=0, where α⩾1, p>1 and λ are real parameters. We prove that all non-trivial solutions are oscillatory and unbounded as x approaches 0. Moreover, there exist γ>0 and δ>0 such that any solution u(x) resembles x−γη(x−δ) near the origin, where η is a non-trivial periodic solution to the Emden–Fowler equation δ2η″+|η|p−1η=0 in [0,∞).

Keywords: oscillatory solutions, singular Sturm–Liouville equation, Emden–Fowler equation

DOI: 10.3233/ASY-151318

Journal: Asymptotic Analysis, vol. 94, no. 3-4, pp. 363-373, 2015