Combined Effects in Singular Elliptic Problems in Punctured Domain

Bachar, Imed; Mâagli, Habib; Eltayeb, Hassan

doi:https://doi.org/10.1155/2021/6630457

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Inhomogeneous Nonlinear Partial Differential Problems: Existence and Non-Existence of Solutions

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Research Article | Open Access

Volume 2021 | Article ID 6630457 | https://doi.org/10.1155/2021/6630457

Combined Effects in Singular Elliptic Problems in Punctured Domain

Imed Bachar,¹Habib Mâagli,^2,3and Hassan Eltayeb¹

Academic Editor: Stanislav Hencl

Received24 Oct 2020

Accepted13 Jan 2021

Published30 Jan 2021

Abstract

The paper deals with nonlinear elliptic differential equations subject to some boundary value conditions in a regular bounded punctured domain. By means of properties of slowly regularly varying functions at zero and the Schauder fixed-point theorem, we establish the existence of a positive continuous solution for the suggested problem. Global estimates on such solution, which could blow up at the origin, are also obtained.

1. Introduction and Statement of Main Results

Consider the problem where be a bounded -domain in containing and satisfying some adequate conditions. In this paper, we are interested in the study of existence and global asymptotic behavior of positive solutions for problem (1). In particular, as it will be seen, the solution may blow up at the origin. The main feature of this paper consists in the presence of the combined effects of singular and sublinear terms in the nonlinearity. Our approach combines properties of Karamata class and Kato class (see [1–4]) with the Schauder fixed-point theorem.

In literature, many researches have studied similar problems for both bounded and unbounded domains (see, for example, [5–13] and the references therein).

In [12], Shi and Yao studied the problem where is a bounded domain in , and is a real parameter The function is required to be in , with in Under various assumptions on they have proved some existence results for belonging to a certain range.

If Colcite and Palmieri [14] showed that (2) has at least one solution provided that and However, if they have also proved that there exists such that (1.2) has a solution for , and no solutions exist if

In [11], by combining monotonicity arguments with variational techniques, Rdulescu and Repovš studied the competition between convex and concave nonlinearity and variable potentials. They have proved that problem (2), with and , has a solution, provided that is small enough.

In [7], Cîrstea et al. investigated the following bifurcation problem: where is a bounded domain, is a nonnegative Hölder function, and is positive, continuous, and nondecreasing function on such that the mapping is nonincreasing in The nonnegative, continuous nonlinearity is assumed to fulfill the hypotheses:

is nonincreasing on with

There exist and so that

They have proved the existence of a unique solution to this problem.

If problem (3) becomes

Many physical phenomena can be described by this kind of problems, known as the Lane–Emden–Fowler equation, see, for example, [7–9, 11, 15] and the references therein.

The study of such problems has been attracted by many researches (see, for instance, [2, 16–25] and the references therein).

In [18], Ghergu et al. considered the semilinear elliptic equation where is the the unit open ball in with , and They have proved that the nonnegative solution of the above equation either has a removable singularity at the origin or it behaves like with

In [25], Zhao et al. investigated the problem where is a bounded smooth domain, and

Using variational techniques, they have proved the existence of infinitely many solutions.

Quoirin and Umezu [20] dealt with the following concave-convex problems under Neumann boundary conditions where is a bounded and smooth domain of and for some

They have proved that is a necessary and sufficient condition for the existence of nontrivial nonnegative solutions of this problem.

In our analysis, we shall extensively use the class of slowly regularly varying functions at zero introduced by Karamata in [26] as follows.

Definition 1. A measurable function , , is said to be slowly varying at zero if is represented in the form where and with

The set of slowly varying at zero (called also Karamata class) is denoted by It is clear that belongs to the class if and only if is a positive function in for some with

Typical examples of slowly varying functions at zero, also used as weight functions (see, [27, 28]), are where is a positive real number, and (-times). For more examples, we refer the reader to [1, 3, 4].

The Karamata class has been frequently used in describing the asymptotic analysis of solutions (see, for examples, [29–33] and the subsequent papers [5, 6, 15, 34–44]).

To simplify our statements in this paper, we need some notations. (i) and denote the Euclidean distance from to (ii) denotes a positive real number such that (iii)For , and defined on such thatdefine the functions and for by (iv) (resp., ) denotes the collection of all (resp., nonnegative) Borel measurable functions in (v)For we say in , if there exists such that for all (vi) denotes Green’s function of the Laplace operator in with Dirichlet conditions(vii)For we set(viii)From ([45], Lemma 9), we know that for any function such that and we have(ix)The letter will denote a positive constant which may vary from line to line

In [46], the authors investigated the problem where and are a positive continuous function in satisfying where , and defined on such that (12) is fulfilled.

They have proved that problem (17) has at least one positive continuous solution on satisfying

In this paper, we aim at generalizing the results obtained in [46] to problem (1).

To this end, we make the following hypothesis:

For is a positive continuous functions in satisfying where , and defined on such that

We may assume that

For we denote by

Define the function on as follows.

If and then

By using properties of slowly regularly varying functions at zero, we prove the following results.

Theorem 2. Under hypothesis , we have for , where

Theorem 3. Assume that hypothesis is satisfied Then, problem (1) admits a solution on satisfying

Remark 4. (i)Theorem 3 generalizes the main result in [46](ii)For we have (iii) where is the function given by (24), (25), (26), and (27).

The paper is organized as follows. In Section 2, we recall some fundamental properties of functions belonging to the Karamata class, and we prove Theorem 2. In Section 3, we prove Theorem 3 by means of the Schauder fixed-point theorem.

2. Karamata Class and Proof of Theorem 2

2.1. Basic Properties of Karamata Class

Lemma 5 (See [1, 3, 4]). If then (i)For every and ,(ii)For every ε>0,

Lemma 6 (Asymptotic behavior, see [1, 3, 4]). If then (i)For every converges and(ii)For every diverges and

Lemma 7 (See [4, 6]). Let then (i) belongs to and(ii)If then belongs to and

2.2. Proof of Theorem 2

The next Proposition, which follows from ([46], Proposition 2.12), will be used.

Proposition 8. Let , and , such that Then for where

We recall that for satisfying where , and defined on such that (21) is fulfilled.

We aim at proving that on where and the function given by (24), (25), (26), and (27)

Throughout the proof, we will apply Lemma 5 and Lemma 7 to verify that some functions are in

We distinguish the following cases:

Case 9. If and then for Therefore, Since and we deduce by Lemma 5 that Using Lemma 6 (i), (21), and Proposition 8 with , and we obtain for Since and we deduce that Now, by using a simple computation, we obtain for Combining this fact with (44), we obtain for

Case 10. If and then for In this case, Since and we deduce that Applying again Proposition 8 with and we obtain for Similarly, by using Proposition 8 with , and we obtain for Hence, Using the fact that and we deduce that Using Lemma 7, ([6], Lemmas 8 and 9) and a simple computation, we deduce that Hence,

Case 11. If and then for Therefore, Since and we deduce that Using Proposition 8 with , and we obtain for Similarly, by using Proposition 8 with , and we obtain for Since and we deduce that By using similar arguments as in the proof of Case 10, we deduce that Hence,

Case 12. If and then In this case, Now, by using Proposition 8 with , and we obtain for By applying again Proposition 8 with , and we obtain for Since and we deduce that As in the proof of Case 11 and Case 10, we deduce that That is The proof is completed.

3. Kato Class and Proof of Theorem 3

3.1. Kato Class

From [24], we recall that Green’s function satisfied

Definition 13. A Borel measurable function in is in the Kato class if Note that this class was introduced in [2] and properly contains the usual Kato class defined (see, [24, 47]) as

Proposition 14 (See [46], Proposition 2.8). For , let and The following properties are equivalent: (i)The function is in (ii) for (iii) or with for

As a consequence of the above Proposition, hypothesis Lemma 5, and Lemma 6 (i), we obtain the following.

Corollary 15. Assume that hypothesis is satisfied and let where the function is given by (24), (25), (26), and (27)
Then, the function

Proposition 16 (See [46], Proposition 2.5). Let with Then, the family is uniformly bounded and equicontinuous in Consequently, is relatively compact in

3.2. Proof of Theorem 3

Assume that hypothesis is satisfied and let be the function given by (24), (25), (26), and (27)

By Theorem 2, there exists such that for each , where is given by (74).

Put and consider the set

By Corollary 15 and Proposition 16, it follows that . So, , closed, bounded, and convex set in

For , set

By using (76), for all we have where which belongs to by Corollary 15. Hence, by Proposition 16, the family of functions is relatively compact in

Next, we aim at proving that

Indeed, by using (76), for all we have

On the other hand,

Hence,

Next, we shall prove the continuity of in the supremum norm. Let be a sequence in which converges to in For each we have

Since from (79), we have and we deduce by Proposition 16 and the dominated convergence theorem that for all

Since is relatively compact in then uniformly. Thus, we have proved that is a compact mapping from to itself.

So, by the Schauder fixed-point theorem, there exists a function such that

Put Then, and satisfies the equation

Since the function and from (85) the function we deduce by (16) that

By (76), we deduce that

Using (87), Remark 4 (iii), and the fact that , we obtain

So, is a solution of problem (1) satisfying (29).

Example 17. Let . Consider such that where with , and Then, by Theorem 3, problem (1) has at least one positive solution satisfying the following estimates: (i)If then for (ii)If then for (iii)If and then for (iv)If then for

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research, King Saud University for funding this research group No. (RG-1435-043). The authors would like to thank the referees for their careful reading of the paper.

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Copyright © 2021 Imed Bachar et al. 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.

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