Existence and Global Asymptotic Behavior of Singular Positive Solutions for Radial Laplacian
The aim of this paper is to establish existence and uniqueness of a positive continuous solution to the following singular nonlinear problem. , where and denotes a nonnegative continuous function that might have the property of being singular at and /or and which satisfies certain condition associated to Karamata class. We emphasize that the nonlinearity might also be singular at , while the solution could blow-up at . Our method is based on the global estimates of potential functions and the Schauder fixed point theorem.
1. Introduction and Main Result
Nonlinear problems of the formwhere is a positive, differentiable function on 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 fluids can be modeled by equations of the form (1) (see for example [11, 12]).
Equations of the form (1) with , appears in a natural manner in those cases when the researcher is looking for radial solutions of Laplace operator.
Let us first introducing the following functional class called Karamata class.
Definition 1. Let and be a function defined on Then belongs to the class if where and with .
Here, it is pertinent to note that the functions in the class are slowly varying, and Karamata developed in  the initial theory in this field.
Cirstea and Rădulescu have exploited in  the Karamata theory to study the asymptotic and qualitative behavior near the boundary of solutions of nonlinear elliptic problems.
The aim of this paper is to address the existence, uniqueness and qualitative behavior of positive continuous solution to the following singular nonlinear problem.where , and denotes a nonnegative continuous function on that might have the property of being singular at and /or and which might satisfies certain condition associated with the Karamata class In this situation, the nonlinearity might also have the property of being singular at Here we emphasizes that the obtained solution may also blow-up at , which is not given in the previous works. Our approach relies on Karamata theory and the Schauder fixed point theorem.
Notations. (i) (resp. ), denotes the set of Borel (resp. nonnegative Borel) measurable functions in
(iii) (resp. ), is the set of continuous (resp. nonnegative continuous) functions in a metric space
(iii) For ,
(iv) For , we say that , , if there exists such that , for all
In what follows, we let , and assume that
(H) withwhere , and , defined on such thatWe introduce the function defined on bywhere andOur main result is the following.
Theorem 2. Let and assume that satisfies Then problem (3) has a unique positive solution satisfying for ,
Remark 3. The solution obtained in Theorem 2 blow-up at for
Example 4. Let and assume that The unique solution of the linear problem is given by Clearly the solution blow up at for and also we have This implies that the global estimates obtained in Theorem 2 are optimal.
2. Karamata Class and Global Estimates
2.1. Karamata Class
It is clear to see that for some , the class is given by Standard examples of functions which are elements of the class are presented below (see [33–35]) where ( times), , and is a sufficiently large positive real number such that is defined and positive on , for some
Next, we collect several properties of the Karamata functions, which will be useful in the proof of our main result.
2.2. Global Estimates
Let , thenis the Green’s function of the operator , with boundary conditions .
Lemma 7. On , we have There exists a constant such that for all ,
For , we define the otential of by Using Lemma 7, we deduce the following.
Corollary 8. Let , then we have
Proposition 9. Let and be a function such that the map , then belongs to and it is the unique solution of the problem
Proof. From Corollary 8, the function is in
Using (21), we have for , Since the function , then and are differentiable on
By simple calculation, we obtain That isBy differentiating (31), we obtain for , Using (31) and the fact that , we obtainFinally, we prove the uniqueness. Let be two solutions of (28) and put Then and Since , we deduce that and therefore Using the fact , we deduce that That is
Proposition 10. Let , and , such thatPut Then for , where and
Proof. For , we have Using (22), we obtain that In what follows, we distinguish two cases.
Case 1 (). In this case So we obtain Since , we deduce that Using Lemma 5 and (34), we deduce that and Hence, it follows by Lemmas 5, 6 and (34) that, for , That isCase 2 (). In this case, Therefore, we have Since , we deduce that Using again Lemma 5 and hypothesis (34), we deduce that and Hence, it follows by Lemmas 5, 6 and hypothesis (34) that, for , we get That isCombining (46) and (52), we obtain for , This ends the proof.
Proposition 11. Assume that condition is satisfied. Then for , we have where
Proof. Let be a function satisfying . Using (5) and (7), we obtain where and
Since and , then one can easy check that and .
Now using Lemmas 5, 6 and Proposition 10 with
and , we deduce that for each , Since and , then we deduce This completes the proof.
3. Existence Results
3.1. Preliminary Results
For , we denote by the following problem Next, we establish several facts to be used within the proof of the main result.
Lemma 12. Let , and be two positive functions satisfyingandThen in
Proof. Let for Assume that for some Then there exists an interval containing such that , on with and or
On the other hand, since , then we have for So So the function is nondecreasing on with
Hence, the function is nondecreasing on with and This gives a contradiction. Therefore,
Proposition 13. Let , and assume that hypothesis is satisfied. Then for each , problem has a unique positive solution satisfying for In particular,
Proof. Let , and assume that the function satisfies hypothesis Note that the functionIndeed, since for each , the function , it follows from (23), (6) and the convergence dominated theorem that
Let and be the closed convex set given by Define the operator on by Since , then for all , we have On the other hand, we have for all Since, for each , the function , then we deduce by (23), (6) and the convergence dominated theorem that is equicontinuous in In particular, for all , and so Moreover, since the family , is uniformly bounded in , then by Ascoli’s theorem that becomes relatively compact in Next, we prove the continuity of in Let and such that as Then we have Now, since we deduce by convergence dominated theorem that Since is relatively compact in , we obtain So is a compact mapping to itself. Therefore, by the Schauder fixed point theorem, there exists such that for each Put , for Then and we haveand We have for all , Now since by hypothesis the function , we deduce from (73) and Proposition 9 that is a solution of problem By Lemma 12, we obtain the uniqueness.
Finally, using (73) and the fact that we deduce that
Corollary 14. Let , and assume that hypothesis is satisfied. For , we denote by the unique positive solution of the problem Then we have
Theorem 15. Let Under hypothesis , problem (3) has a unique positive solution satisfying for
Proof. Let be a sequence that decreases to zero. Let be the unique positive continuous solution of the problem By Lemma 12 (or Corollary 14), the sequence decreases to a function , and from (73) the sequence increases to Using this fact and (73), we obtain for each , where and is given by (63).
By the monotone convergence theorem, we obtain