DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

PICONE-TYPE INEQUALITIES FOR A CLASS
OF QUASILINEAR ELLIPTIC EQUATIONS
AND THEIR APPLICATIONS
NORIO YOSHIDA
Picone-type inequalities are established for quasilinear elliptic equations with first-order
terms, and oscillation results are obtained for forced superlinear elliptic equations and
superlinear-sublinear elliptic equations.
Copyright © 2006 Norio Yoshida. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
There is an increasing interest in oscillation problems for half-linear differential equations.
There are many papers dealing with half-linear partial differential equations (see,
e.g., Bognár and Došlý[1], DošlýandMařík [2], Dunninger [3], Kusano et al. [6], Mařík
[7], Yoshida [8], and the references cited therein). Superlinear elliptic equations with
p-Laplacian principal part and superlinear-sublinear elliptic equations were studied by
Jaroš etal.[4, 5]. Picone identity or inequality plays an important role in establishing
Sturmian comparison and oscillation theorems for partial differential equations.
The objective of this paper is to establish Picone-type inequalities for quasilinear partial
differential operators P and ˜P defined by
P[v] ≡∇·(A(x)|∇v| α−1 ∇v ) +(α +1)|∇v| α−1 B(x) ·∇v + C(x)|v| β−1 v,
˜P[v] ≡∇·(A(x)|∇v| α−1 ∇v ) +(α +1)|∇v| α−1 B(x) ·∇v
(1.1)
+ C(x)|v| β−1 v + D(x)|v| γ−1 v,
where α, β, andγ are constants satisfying α>0, β>α,0