Existence and asymptotic behavior of positive solutions of fourth order quasilinear differential equations. (English) Zbl 1293.34063

Kusano, Takasi; Manojlovic, Jelena; Tanigawa, Tomoyuki

Taiwanese J. Math. 17, No. 3, 999-1030 (2013).

The authors study the existence and the precise asymptotic behavior of solutions \(x(t)\) defined on \([a,\infty )\) of the sub-half-linear (i.e., \( \alpha >\beta \)) equation \[ (|x''|^{\alpha -1} x'')''+q(t)|x|^{\beta -1} = 0 \] of Emden-Fowler type (i.e., \(q:[a,\infty )\rightarrow (0,\infty )\)).
In the papers by F. Wu [Funkc. Ekvacioj, Ser. Int. 45, No. 1, 71–88 (2002; Zbl 1157.34319)] and by M. Naito and F. Wu [Acta Math. Hung. 102, No. 3, 177–202 (2004; Zbl 1048.34077)], a classification of (positive) \(x(t)\) was presented and the existence and the precise asymptotic behavior of solutions belonging to some of the classes was proved for both cases \(\alpha >\beta \), \(\alpha <\beta \).
Here, by assuming that \(q\) is regularly varying in the sense of Karamata, it is proved that the solutions of all remaining classes also exist and their precise behavior is determined for the sub-linear case \(\alpha >\beta \).

Reviewer: Vojislav Marić (Novi Sad)

MSC:

34D05	Asymptotic properties of solutions to ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations
26A12	Rate of growth of functions, orders of infinity, slowly varying functions

Keywords:

fourth-order differential equations; regularly varying solutions; asymptotic behavior of solutions

Citations:

Zbl 1157.34319; Zbl 1048.34077