Oscillation constants for second-order ordinary differential equations related to elliptic equations with p-Laplacian

• Published in: Nonlinear analysis Vol. 113; pp. 115 - 136
• Main Authors: Došlý, Ondřej, Yamaoka, Naoto
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.01.2015
• Subjects: [formula omitted]-Laplacian; Complement; Constants; Differential equations; Half-linear differential equations; Mathematical analysis; Nonlinearity; Oscillation; Oscillation constant; Oscillations; Phase plane analysis; Riccati technique; Tantalum; Time maps; Phase plane analysis; p-Laplacian; Oscillation constant; Oscillation; Half-linear differential equations; Riccati technique; Time maps
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2014.09.025

In this paper we consider the second-order nonlinear differential equation (∗)(tα−1Φ(x′))′+tα−1−pf(x)=0,Φ(x)=|x|p−2x,p>1,α∈R, with f satisfying xf(x)>0, x≠0. We analyze the difference between the cases α<p, α>p, and α=p. In each case we give a condition on the function f which guarantees that solutions of Eq. (∗) are (non)oscillatory. The principal methods used in this paper are the Riccati technique and its modifications. The results of our paper complement and extend several previously obtained results on the subject.