DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

POSITIVE SOLUTIONS OF SECOND-ORDER DIFFERENTIAL
EQUATIONS WITH PRESCRIBED BEHAVIOR OF
THE FIRST DERIVATIVE
OCTAVIAN G. MUSTAFA AND YURI V. ROGOVCHENKO
Using Banach contraction principle, we establish global existence of solutions to the
nonlinear differential equation u ′′ + f (t,u,u ′ ) = 0thathaveasymptoticdevelopments
u(t) = c + o(1) and u(t) = ct + o(t d )ast → +∞ for some c>0andd ∈ (0,1). Our theorems
complement and improve recent results reported in the literature. As a byproduct,
we derive a multiplicity result for a large class of quasilinear elliptic equations in exterior
domains in R n , n ≥ 3.
Copyright © 2006 O. G. Mustafa and Y. V. Rogovchenko. 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
The quasilinear elliptic equation
Δu + f (x,u)+g ( |x| ) x ·∇u = 0, |x| >A>0, (1.1)
describes several important phenomena arising in mathematical physics. Equation (1.1)
has been investigated recently in [1, 2, 4, 15, 16], where existence of eventually positive
and decaying-to-zero solutions was discussed by using approaches based on Banach
contraction principle and exponentially weighted metrics, sub- and supersolutions, and
variational techniques.
Let G A ={x ∈ R n : |x| >A}, n ≥ 3. Similarly to [14], we assume that the function
f : G A × R → R is locally Hölder continuous and g :[A,+∞) → R is continuously differentiable.
Following [2, 4], we also suppose that f satisfies
0 ≤ f (x,t) ≤ a ( |x| ) w(t), t ∈ [0,+∞), x ∈ G A , (1.2)
where a :[A,+∞) → [0,+∞), w :[0,+∞) → [0,+∞) are continuous functions,
for certain M, ε>0, and g takes on only nonnegative values.
w(t) ≤ Mt, t ∈ [0,ε], (1.3)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 835–842