Research Article
Existence and Global Asymptotic Behavior of Singular Positive
Solutions for Radial Laplacian
Imed Bachar ,
1
Habib Mâagli ,
2
and Said Mesloub
1
1
King Saud University, College of Science, Mathematics Department, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2
King Abdulaziz University, College of Sciences and Arts, Rabigh Campus, Department of Mathematics, P. O. Box 344,
Rabigh 21911, Saudi Arabia
Correspondence should be addressed to Imed Bachar; abachar@ksu.edu.sa
Received 2 December 2018; Accepted 2 January 2019; Published 3 February 2019
Academic Editor: Mark A. McKibben
Copyright © 2019 Imed Bachar et al. Tis 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.
Te aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear
problem. {−
1−
(
−1
)
= ()
, ∈ (0, 1), lim
→0
−1
() = 0, (1) = 0}, where ≥ 3, < 1, and denotes a nonnegative
continuous function that might have the property of being singular at =0 and /or =1 and which satisfes certain condition
associated to Karamata class. We emphasize that the nonlinearity might also be singular at =0, while the solution could blow-up
at 0. Our method is based on the global estimates of potential functions and the Schauder fxed point theorem.
1. Introduction and Main Result
Nonlinear problems of the form
−
1
(
)
=(,), ∈(0,1),
> 0, in (0,1),
(1)
where is a positive, diferentiable function on (0,1) and
satisfying several suitable conditions have been studied by
many researchers (see for instance [1–10]). Note that many
problems in the boundary layer theory and the theory of
pseudoplastic fuids can be modeled by equations of the form
(1) (see for example [11, 12]).
Equations of the form (1) with () =
−1
( ≥
3), appears in a natural manner in those cases when the
researcher is looking for radial solutions of Laplace operator.
For a multiple results of existence, uniqueness, and
asymptotic behavior associated with similar problems, we
refer the reader to [13–30] and their bibliographies.
Let us frst introducing the following functional class K
called Karamata class.
Defnition 1. Let >1 and be a function defned on (0,].
Ten belongs to the class K if
() fl exp (∫
()
), (2)
where >0 and ∈ ([0, ]) with (0) = 0.
Here, it is pertinent to note that the functions in the class
K are slowly varying, and Karamata developed in [31] the
initial theory in this feld.
Cirstea and R˘ adulescu have exploited in [32] the Kara-
mata theory to study the asymptotic and qualitative behavior
near the boundary of solutions of nonlinear elliptic problems.
Te aim of this paper is to address the existence, unique-
ness and qualitative behavior of positive continuous solution
to the following singular nonlinear problem.
−
1−
(
−1
)
=()
, ∈(0,1),
lim
→0
−1
() = 0, (1) = 0,
(3)
where ≥3, <1 and denotes a nonnegative continuous
function on (0,1) that might have the property of being
singular at =0 and /or =1 and which might satisfes
Hindawi
Journal of Function Spaces
Volume 2019, Article ID 3572132, 9 pages
https://doi.org/10.1155/2019/3572132