Abstract
In the present paper, in view of the variational approach, we discuss the Neumann problems with -Laplacian-like operator and nonstandard growth condition, originated from a capillary phenomena. By using the least action principle and fountain theorem, we prove the existence and multiplicity of solutions to the class of Neumann problems under suitable assumptions.
1. Introduction and Main Result
In this paper, we study existence and multiplicity of solutions for the -Kirchhoff-type equation involving nonstandard growth condition and arising from a capillary phenomena of the following type: where is a bounded domain with smooth boundary , , , and ; is the outward unit normal on . with It is worth mentioning that in problem (1) contains the Kirchhoff-type functions such as , , and . The Kirchhoff-type equation has a strong physical background. We refer the interested readers to [1–3] and the references therein.
It is well-known that is called -Laplacian operator. The study of differential equations and variational problems with nonstandard -growth conditions and -Laplacian operator (see [4, 5]) has been a very interesting topic in recent years. They allow the modelling of various phenomena that arise in the study of elastic mechanics and image restoration, and we can refer to [6–8]. Since the left-hand side of problem (1) contains an integral over , it is no longer a pointwise identity. Therefore, it is often called a nonlocal problem. Problem (1) has a rich mathematical and physical background base. For example, when , , problem (1) degenerates into a generalized capillarity equation describing “capillary phenomena.”
The problem involving -Laplacian-like operator was firstly studied by Rodrigues based on the variational method (see [9]). He studied the following equation: where is a bounded domain with smooth boundary , , and . The author proved existence and multiplicity of solutions by using mountain pass theorem and fountain theorem. Problem (3) can be used to describe capillarity. Capillarity can be briefly explained by considering the effects of two opposing forces: adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those of the container; and cohesion, i.e., the attractive force between the molecules of the liquid. The study of capillary phenomena has gained some attention recently. This increasing interest is motivated not only by fascination in naturally occurring phenomena such as motion of drops, bubbles, and waves but also its importance in applied fields ranging from industrial and biomedical and pharmaceutical to microfluidic systems. In reference [9–14], the existence and multiplicity of solutions for the Dirichlet boundary value problems with -Laplacian-like operator are also studied.
In recent years, the Neumann problems have been extensively studied, and we can refer to [15–21]. The authors in [20, 21] studied the following -Kirchhoff-type Neumann problems: where is a bounded domain with smooth boundary and is the outward unit normal on . By using the saddle point theorem and abstract linking argument due to Brezis and Nirenberg [22], they showed that problem (4) has at least two nontrivial solutions. In [16], Chung have extended problem (4) to -Kirchhoff-type Neumann problems. Under appropriate assumptions on , the author proved the existence and multiplicity of solutions by using abstract linking argument. From a mathematical point of view, the extension from the -Laplace operator to the -Laplace operator is interesting and not trivial, since the -Laplace operator has a more complicated structure than the -Laplace operator, for example, they are nonhomogeneous.
Furthermore, Jiang et al. [17] have studied the existence and multiplicity of solutions for problem (4) of the Kirchhoff function under the Landesman-Lazer type condition by using saddle point theorem and abstract linking argument. Also for further studies on this Landesman-Lazer type condition, we have provided reference for readers in [23–26].
However, we find that the -Laplacian-like Neumann problems with Landesman-Lazer type condition is lagged behind. We point out that the main difficulty arises from the fact that the first eigenvalue of the -Laplacian is not isolated. In this case, we can use the technique of decomposing the space in this paper.
Motivated by the above papers and the results, the main purpose of this paper is to study the existence and multiplicity of solutions for problem (1). Assuming that the following conditions are met.
() There is a constant , such that , for all .
() There is a constant , such that where .
() , and there exists a constant such that where .
() uniformly for a.e. , where .
() uniformly for a.e. .
() (Landesman-Lazer type condition) Whenever is such that and as , then
() uniformly for a.e. .
() for all .
The main results of this paper are as follows.
Theorem 1. Suppose that . If , , , and () hold, the problem (1) has at least one solution.
Theorem 2. Suppose that . If , , , , , , and hold, the problem (1) has infinitely many nontrivial solutions.
In the present paper, first of all, we use the least action principle method to study the existence of solution for the problem (1). Furthermore, we give some new solvability results for the problem (1) under the Landesman-Lazer type condition. By imposing additional assumptions on , we establish the existence of infinitely many solutions by using fountain theorem, due to Bartsch in [27].
2. Preliminaries
We start with some preliminary basic results on the theory of Lebesgue-Sobolev spaces and with variable exponent; for more details, we refer the reader to the book by Musielak [28] and Fan and Zhao [29].
Let denoted the set of all measurable real functions defined on . For any , the variable exponent Lebesgue space is defined as with the norm
The variable exponent Sobolev space is defined as with the norm
Proposition 3 (see [29]). (i)For any , with , the inequality holds as follows(ii)If and (respectively ) for any , then is embedded continuously (respectively, compactly) in
Let us now recall the modular function which plays an important role in the variable order Lebesgue spaces and which is defined by
Proposition 4 (see [30]). For any , then the following properties hold:
We can split in the following way. Let
For each , denote where , and . Note is a closed linear subspace of with codimension 1. Then, (see [31]).
The following proposition plays an important role in our proof.
Proposition 5. (see [31], Proposition 2.6) there is a positive constant , such that
We define the norm by
We conclude that and are equivalent norms. Invoking Proposition 5, it is easy to see that and are equivalent norms in .
The modular function define by
A similar derivation of [29] has the following proposition.
Proposition 6. Set . Then, the following properties hold the following:
Proposition 7 (see [9, 30]). and its derivative is given by For all and the following properties hold the following: (i) is convex and sequentially weakly lower semicontinuous(ii) is a mapping of type , that is, if in and imply in (iii) is a strictly monotone operator and homeomorphism
Definition 8. We say that is a weak solution of problem (1), and if for any , it satisfies the following
The energy functional associated to problem (1) is defined as where . Obviously, functional and for all . It is well-known that the weak solutions of problem (1) correspond to the critical point of the functional on .
Definition 9. Let be a Banach space and . We say that satisfies the condition if any sequence such that and in as has a convergent subsequence.
Let is a reflexive and separable Banach space, and then, there exists and such that
For convenience, we write
Lemma 10 (see [32] fountain theorem). Let be a Banach space, is an even functional and satisfies the condition. If for every , there exist such that
(): ;
(): .
Then, has a sequence of critical values tending to .
3. The Proof of Theorem 1.1
In this section, the existence of at least one solution for problem (1) is obtained in . We start with one auxiliary result.
Lemma 11 (see [29]). By Sobolev’s inequality, there exists a positive constant such that for all .
Proof of Theorem 1.1. The proof is divided into two steps as follows.
Firstly, we show that is coercive. By and , there exists a constant and , which is subadditive, that is
for all , and coercive, so
as , and satisfies that
for all , such as
for all and .
In fact, since as uniformly for all , there exists a sequence of positive integers with for all positive integer such that
for all and all . Let and define
for , where .
By the definition of , we have
for . By , one has
for all and . It follows that
for all and by (33), where
with and . In fact, when , we have
for all and .
It is obvious that is continuous and coercive. Moreover, one has
for all . In fact, for every , there exists such that
which implies that
for all by (33) and the fact that for all integers .
Now, we only need to prove the subadditivity of . Let
and . Then, we have
Hence, by (33), we obtain
which shows that is subadditive.
Let , . So . By conditions , , (16), (26)–(30), and Proposition 6, we have
Note that , we have . Since is coercive and , which implies that is coercive.
Next, we show that the is weakly lower semicontinuous. Let
Since as , for a.e. . By Fatou Lemma, we have
Then
So, the is weakly lower semicontinuous.
Hence, by the least action principle (see [17]), has a minimum. So problem (1) has at least one solution in , which completes our proof.
4. The Proof of Theorem 1.2
In this section, we will show that problem (1) has infinitely many nontrivial solutions by using Lemma 10. Firstly, we need to prove lemma as follows.
Lemma 12. Suppose that the conditions of Theorem 2 hold. Then, satisfies the condition.
Proof. Let be a sequence such that is bounded and as . Firstly, we prove that is bounded in . Arguing by contradiction, we assume that any subsequence of (still denoted by ), we have as . Let . Note that and are bounded in . Now, we can find and a subsequence of (still denoted by ), with
By conditions and , we have
By , for any , there is a constant such that
There exists a positive constant such that
Then, by (51), one deduces that
In view of (52) and the fact that as , one has
Let , we have
On the other hand, by the weakly lower semicontinuous of the norm, one has
then
From (54) and (56), we derive that
Therefore, for all which yields . It follows that and hence that
which means that as . By condition and as , one deduces
Since as , we have
which contradicts to (59). Therefore, we conclude that the sequence is bounded in .
Secondly, we will prove that has a convergent subsequence in . Since is bounded, there exist a subsequence, still denoted by and such that
By (12), (61), and , we have
as . One has
According to , we have . Therefore
as . By (63) and condition , we can deduce from that
as . Invoking the condition (see Proposition 7), we can deduce that strongly in as . So, satisfies the condition. The proof is complete.
The following lemma plays an important role in our proof.
Lemma 13 (see [30]). If , for any , denote then .
Lemma 14. Suppose that the conditions of Theorem 2 hold. Then, there exist such that
(): ;
(): .
Proof. To prove . In view of , we obtain
for all and . Furthermore, for any with , it follows from conditions , , (26), and Proposition 4, and we have
Choose . For any with , we infer that
Note that and as , we assert as . We have
as . Which implies .
Next, we will prove . Let
by condition , we have
is monotonically decreasing with respect to the variable as . So , we have for any , and we have
where ,
By conditions and , for any , there exists , and we have
By (73), (75), and Proposition 4, for any with large enough, we have
Since and equivalence of norm in finite dimensional space, it is clear that when large enough. Therefore, we conclude
The proof is complete.
Proof of Theorem 1.2. Since is also a reflexive and separable Banach space, we can give the decomposition to as (25). According to and Lemma 12, we deduce that is an even functional and satisfies condition in . By Lemma 14, it was proven that if is large enough, there exist such that and hold in . Thus, we can deduce that satisfies all conditions of Lemma 10 in . Therefore, has an unbounded sequence of critical values, which implies that problem (1) has infinitely many nontrivial critical points in . This completes the Proof of Theorem 1.2.
Data Availability
No underlying data was collected or produced in this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 11861021).