Existence Results for a Class of the Quasilinear Elliptic Equations with the Logarithmic Nonlinearity

Abstract

In this paper, the nonlinear quasilinear elliptic problem with the logarithmic nonlinearity in was studied. By means of a double perturbation argument and Nehari manifold, the authors obtain the existence results.

1. Introduction

In this paper, we consider the existence of solution to the following problem where , , and . We always suppose that is a sign-changing function; is a function.

Equations of the above form are mathematical models occurring in studies of the -Laplace equation, generalized reaction-diffusion theory [1], non-Newtonian fluid theory [2, 3], non-Newtonian filtration theory [4, 5], and the turbulent flow of a gas in porous medium [6]. In the non-Newtonian fluid theory, the pair is a characteristic quantity of the medium. Media with are called dilatant fluids, and those with are called pseudoplastics. If , they are Newtonian fluids. When , the problem becomes more complicated since certain nice properties inherent to the case seem to be lose or at least difficult to verify. The main differences between and can be founded in [7, 8].

In recent years, logarithmic nonlinearity is widely used in pseudo-parabolic equations which describe the mathematical and physical phenomena. Equations of the type have been studied by many authors when (see, for example, [9–12] and the reference therein). To do so, the authors always use the nice properties of , such that, maximum principle and comparison principle and so on. Meanwhile, existence and structure of solutions for such equations with in bounded domains have also attracted much interest (see [13, 14]).

In the following discussion, we consider two different situations. Firstly, we consider the existence of positive solution for problem with Neumann boundary conditions. In this case, suppose that are also radial functions, , in . Our strategy in the study of problem is to adopt a double perturbation argument. First, following [15, 16] (see also [17]), for each , we consider a family of approximate problems

Then, it is natural to look for a family of solutions of and then to pass the limit as to obtain a solution to .

For each , define . Consider the second family of problems

Here, is an appropriate constant. When , we get a solution to . The role of problem is that we cannot use Poincare inequality to solve directly by variational methods.

Secondly, we consider the multiple solutions for problem with Dirichlet boundary conditions. In this case, we consider is a sign-changing function, . The method is based on Nehari manifold and logarithmic Sobolev inequality.

By modification of the methods given in [18–22], we obtain the following results.

Theorem 1. Let be any radial function. Then, problem has a positive radial solution .

Remark 2. Theorem 1 is valid even if we change the logarithm by a more general singular function. In fact, suppose is a smooth function such that for some . Then, we can perturb by a family of smooth functions decreasing in , such that and pointwise in as . This perturbation can be done in such a way that for some , and then, all the results in Section 2 hold with little modification.

Theorem 3. Let and changes sign in , satisfying where is the volume of in . Then, possesses at least two nontrivial solutions.

The paper is organized as follows. In Section 2, we construct a sub- and a supersolution for and finish the proof of Theorem 1. In Section 3, we prove Theorem 3 by the method of Nehari manifold and logarithmic Sobolev inequality.

2. Proof of Theorem 1

2.1. Sub- and Supersolution for

Lemma 4. Suppose that . Then, the function is a subsolution for which does not depend on and .

Proof. We just need to see that, since in , the following inequality holds independently of and :

We proceed to find a supersolution for . Denote by , the following subspace of :

For , we define the expression:

Remark. The expression defines a norm on , and is a reflexive Banach space. Furthermore, by ([23], (7.44)), the Poincare inequality holds on , that is, there exists such that

Next, we work with the radial formulation for in the specific case that , where . Notice that

For simplicity, denote

Then, if solves we will have that is a solution of Eq. (10). In order to prove existence of such , we find a minimum of the functional in the sequel. Let denote the set of symmetric functions with respect to the origin. We define by where and .

Lemma 5. The functional is , weakly lower semicontinuous and coercive so that there exist such that

The proof is standard by (9). Also, since is a weak solution of (13), we have in which

Then, we define

Lemma 6. Suppose that . Then, the function is a supersolution for which does not depend on .

Lemma 7. There exists a constant such that and the constant does not depend on . Moreover, for each , there exist a constant and such that we have the following estimates:

Proof or Lemmas 6 and 7 can be found in [18], we omit them here.

2.2. Existence of Solution for

In this section, we use the sub- and supersolution from Section 2.1 ( and , respectively) to obtain a solution for the problem . Define the function where we choose in such a way that the function is increasing in for all . Now, starting with , we define a sequence such that each satisfies

Let us now recall Lemma 2.1 in [24],

Lemma 8 (weak comparison principle). Let be a bounded domain in with smooth boundary and is continuous and nondecreasing, let satisfy

For all nonnegative . Then, the inequality implies that

Lemma 9. The sequence is nondecreasing and satisfies for all and all .

Proof. We just need to see that and the general case follows by induction in an analogous way. We have

So, we can apply Lemma 8 and obtain that in . On the other hand,

Again, Lemma 8 implies in .

By Lemma 9, we define the pointwise limit and we see that

The function is in fact a solution of .

Lemma 10. The function is a solution of , and it belongs to .

Proof. For each , we have

Since we have we obtain, as in Lemma 7, that is bounded. Then, for a subsequence that we still denote by , we have the convergence

2.3. Obtaining a Solution for

In this section, we pass the limit as and then obtain a solution for .

Lemma 11. For a fixed , the problem has a solution which is obtained as the limit of as .

Proof. For simplicity, we omit the dependence on for . We know that

Also, we have

As in Lemma 7, we can prove, for each , there exist a constant and such that we have the following estimates:

Then, from the compact imbedding , we see that there exist a sequence and defined on such that, if we define , then

2.4. Concluding the Proof of Theorem 1

Now, we would like to pass the limit in the family obtained in Section 2.3 and get a solution to . In order to do that, we need some estimates like the ones in Lemma 7 independently of .

First, we observe that the following estimate holds in

Notice that the family satisfies for . We know that if ,and if .

From Eqs. (36)–(38) we see, as in Lemma 7 that, for each for each , there exist a constant and such that we have the following estimates:

Now, arguing as in Section 2.4, we can find a function which satisfies that is, is a radial solution for the problem .

We see that . Now, extend continuously to the whole interval . Indeed, let be a sequence in with as . From Eq. (13) (after we have passed the limit in )

Then, if , we get

From Eq. (36), we obtain that there exists a constant such that so is a Cauchy sequence in . Let be the limit of such sequence. By a similar argument, we conclude that if is another sequence in converging to 0, then we necessarily have . So, we have proved that finishing the proof of Theorem 1.

3. Proof of Theorem 3

3.1. Preliminaries

In this section, we consider the multiple solutions for problem with Dirichlet boundary conditions. In this case, we consider is a sign-changing function, . Moreover, it is necessary to note that the presence of the logarithmic nonlinearity leads to some difficulties in deploying the potential well method. In order to handle this situation, we need the following logarithmic Sobolev inequality which was introduced by [25].

Proposition 12. Let , , and . Then, we have where

Remark. If then, by defining for , we derive for any real number .

We start by considering the energy functional by in which .

Lemma 13. For and , let , then it holds in which .

Proof. Using the fact , we have Let , then where By direct calculations, we know Also, by logarithmic Sobolev inequality (47) and (51), we have Then, combining (50), (51), (53), and (54), we have Taking in (55), then we know (49).

Lemma 14. [19] Let be a sequence in . If and in , then

If in , then

3.2. Multiple Solutions

Inspired by [19], we seek the weak solutions of by Nehari manifold. First, a simple calculation shows that , and its derivative is given by for all .

From (49), is not bounded on , but we can prove that is bounded from below on Nehari manifold where denotes the usual duality.

It is clear that all nontrivial critical points of must lie on , and as we will see below, local minimizers on are usually critical points of . Also, we can see that

Let and consider the real function for defined by

Such maps are known as fibering maps which were introduced by Drabek and Pohozaev [26].

Then, by direct calculations, we have

Lemma 15. Let and . Then, if and only if .

Proof. First, by direct calculations, we know Since , then if and only if .

Then, if , we have and

Thus, we can divide into three subsets , , and , where

Lemma 16. If is a local minimizer for on and . Then, .

Proof. If is a local minimizer for on , by Lagrange multipliers, there exists such that where .
Since , then On the other hand, from , we can see Then, and .

Proposition 17. Both and are nonempty.

Proof. From (62), has a unique turning point at Since is sign-changing, then we can take such that Also, we can take such that Then, both and are nonempty.

Just like [19], by Lemmas 13–16, we can get the following results.

Lemma 18. [19] is bounded; is bounded below on .

Lemma 19. [19] Every minimizing sequence for on is bounded, , .

Proposition 20. has a minimizer on .

Proof. Let be a minimizing sequence, i.e., .
By Lemma 18, is bounded; we may assume that Since , we can get Suppose in , then Then, there exists such that , and then, attains minimum at .
Hence which is impossible. Hence, in , , and , this means that is a minimizer for on .

Proposition 21. There exists a minimizer of on .

Proof. Let be a minimizing sequence. By Lemma 19, is bounded; we may assume that Since , by Lemma 19, we can get Suppose in , then Then, there exists such that , , but in .
Hence Since the map attains its maximum at , This means is impossible.
Hence, in , and this means that is a minimizer for on .

Proof of Theorem 3. Propositions 20 and 21 show that the energy functional has two minimizers on and on . Next, by Lemma 16, has two critical points and on , which means that the problem has at least two nontrivial solutions under the condition .

Data Availability

All the data in our manuscript are available.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contributions

All authors carried out the proof and conceived of the study. All authors read and approved the final manuscript.

Acknowledgments

The project is supported by the Foundation for Basic Subject of Army Engineering University of PLA (KYJBJQZL1924), the Major Projects of Natural Sciences of University in Jiangsu Province of China (No. 18KJA110003), the MOE (Ministry of Education in China) Youth project of Humanities and Social Sciences (No. 16YJC630186), and the Qianfan Plan (Mathematical Modeling and Application Team) of Jiangsu Maritime Institute (No. 014060).