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1 Introduction and main results

In this paper, we are interested in the existence of least energy nodal solutions for the following Kirchhoff-type variable-order fractional Laplacian problems with critical variable:

growth:

where

is a continuous function, is a bounded domain in with Lipschitz boundary, is a parameter, is the variable-order fractional Laplace operator, for all The variable-order fractional Laplace operator is defined as follows:

along any where denotes the Cauchy principle value. the variable-order fractional Laplace operator reduces to the usual fractional Laplace operator; see [14,15] for the concise introduction to the fractional Laplace operator and related variational results.

We now impose the assumptions on the functions and that will in full force throughout the paper. Firstly, we suppose that is a continuous function satisfying the following assumptions:

From now on, for the variable exponents we set

Moreover, we suppose that satisfies the following conditions:

(f1)

(f2) for all . ∈ R and all . ∈ .;

(f3)

A typical example of function fulfilling hypotheses is as follows:

The main driving force for studying problem (1) includes two aspects. On the one hand, when = 1 Eq. (1) reduces to the general Kirchhoff-type model. Recently, some researchers also explored such equations in the study of nonlinear vibrations theoretically or experimentally. For example, Carrier [1] used a more rigorous method to deduce a more general Kirchhoff model. Moreover, the nonlocal Kirchhoff problems of parabolic type can model several biological systems such as population density; see, for instance, [4]. In fact, the energy functionals of (1) have obviously different properties from the case , and thus, several mathematical difficulties arise naturally in the study of the case by variational and topological methods. It is worth mentioning that Fiscella and Valdinoci [9] deduced a new Kirchhoff model involving the fractional Laplacian by considering the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string; see [9, App. A] for more details.

In recent years, finding sign-changing solutions to the Kirchhoff-type problems has been an attractive subject, and many interesting results have been obtained. In the following, let us sketch some advances related to the subject of our paper. Concerning the advances of Kirchhoff-type problems in the bounded domains, Zhang and Perera in [26] applied the method of invariant sets of descent flow to investigate the existence of signchanging solutions for Kirchhoff-type problems; see also Mao and Zhang in [12] for more related results via similar approaches. Using the constraint variational methods, Shuai in [18] obtained that Kirchhoff-type problems has one least energy sign-changing solution and the energy of strictly larger than the ground state energy. After that, with the help of non-Nehari manifold method, Tang and Cheng in [19] generalized some results obtained in [18]; see also [2] for more general Kirchhoff-type function in this direction. In [20], Wang obtained the following results for Kirchhoff-type equation with critical growth by employing the constraint variational method and the quantitative deformation lemma: the existence of least energy sign-changing solutions u. and the energy of u. is strictly larger than twice that of the ground state solutions. Concerning the advances in the abstract Kirchhoff framework, here we just review two papers as follows: by using the minimization argument and a quantitative deformation lemma, Figueiredo et al. in [7] investigated the existence of a sign-changing solution for the following Kirchhoff-type equation:

where . is a bounded domain in

where is a bounded domain in is a continuous function with some appropriate assumptions, and is a superlinear class function with subcritical growth. In unbounded domains, Figueiredo and Santos Júnior in [8] obtained a least energy signchanging solution to a class of nonlocal Schrödinger–Kirchhof problems involving only continuous functions by using a minimization argument and a quantitative deformation lemma. Moreover, the authors also proved that the problem has infinitely many nontrivial solutions when it presents symmetry. In [3], Cheng and Gao studied the following Kirchhoff-type problem, which involves a fractional Laplacian operator:

where is a constant, satisfies subcritical growth. The authors proved the existence of least energy sign-changing solutions for this problem by using the constraint variation method and quantitative deformation lemma.

On the other hand, variable-order fractional Laplacian problems was introduced by Xiang et al. in [25]. They studied the following variable-order fractional Laplacian problems involving variable exponents:

Under some suitable assumptions, they showed that problem (2) admits at least two distinct solutions by applying the mountain pass theorem and Ekeland’s variational principle. Subsequently, Wang and Zhang in [21] proved the existence of infinitely many solutions for possibly degenerate Kirchhoff-type variable-order fractional Laplacian problems by using the new version of Clark’s theorem due to Liu and Wang in [11]. Very recently, Xiang et al. in [24] obtained the existence of two solutions for a class of degenerate Kirchhoff-type variable-order fractional Laplacian problems by employing the Nehari manifold approach.

However, regarding the existence of sign-changing solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents, there has been no paper in the literature as far as we know. Hence, a natural question is whether or not there exists sign-changing solutions of problem (1)? Another interesting question is whether or not the same conclusion still holds for critical exponent The goal of the present paper is to give an affirmative answer.

The corresponding energy functional to problem (1) is defined by

It is standard to verify that belongs to , and the critical points of are the solutions of problem (1). Furthermore, if we write and then every solution problem (1) with the property that is a sign-changing solution of problem (1).

Now, we give the following first main result.

Theorem 1. Assume that (s1).(s2) and (f1).(f3) hold. Then there exists such that for all problem (1) has a least energy sign-changing solution

Another objective of this paper is to establish the so-called energy doubling property (cf. [22]), i.e., the energy of any sign-changing solution of problem (1) is strictly bigger than twice that of the ground state solution. We have the following result.

Theorem 2. Assume that (s1).(s2) and (f1).(f3) hold. Then there exists such that for all is achieved and where and u is the least energy sign-changing solution obtained in Theorem 1. In particular, is achieved either by a positive or a negative function.

Remark 1. As an application of Theorem 2, problem (1) with and

has a least energy sign-changing solution . with energy doubling property.

2 Preliminaries

In this section, we first recall some definitions and results of variable exponent Lebesgue spaces (see [5,6,16]), which will be used later.

be a nonempty open set. A measurable function is named a variable exponent.

The variable exponent Lebesgue space is

with the Luxemburg norm

then is a Banach space, and when is bounded, we have the following relations:

That is, if is bounded, then norm convergence is equivalent to convergence with respect to the modular For the bounded exponent, the dual space can be identified with where the conjugate exponent is defined by then the variable exponent Lebesgue space is separable and reflexive. So we can see that Hölder’s inequality is still valid in the variable exponent Lebesgue space. For all the following inequality holds:

Next, we give some definitions and results of variable-order fractional Sobolev spaces.

Define as the linear space of Lebesgue measurable functions from such that any function

Equip with the norm

Similar to the proof of Lemma 7 in [17], we can show that is a Hilbert space. In this paper, we used norm to study problem (1).

Lemma 1. (See [25, Lemma 2.1].) The embeddings are continuous. Moreover, if for any fixed constant exponent can be continuously embedded into

The following embedding theorem shows that the variable-order fractional Sobolev space is related with the variable exponent Sobolev spaces.

Lemma 2. be a smooth bounded domain. Assume that s : are two continuous functions satisfying respectively. Then there exists such that for any it holds that

That is, the embedding is continuous. Furthermore, this embedding is compact. If , then there exists such that

3 Some technical lemmas

Now, for fixed we define function and mapping by

and

Lemma 3. Assume that (s1).(s2) and (f1).(f3) hold. If then has the following properties:

(i)The pair is a critical point of with if and only if

(ii) The function has a unique critical point which is also the unique maximum point of on Furthermore, if

Proof. (i) By definition of we have that

Thus, item (i) holds

(ii) For any

From (f1) and (f2), for any there is such that

Then by the Sobolev embedding theorem we have

Choose such that . we have that for small enough and all Similarly, we are also able to prove that for small enough and all Therefore, there exists such that

On the other hand, by (f2) and (f3) we claim

Therefore, choose is large enough, it follows that

Similarly, we have be large enough, we obtain

for all Combining (6) and (8) with Miranda’s theorem [13], there exists such that T.

According to the proof in [20], we can prove the uniqueness of the pair

Lastly, we prove that

Suppose we have

On the other hand, by we have

So, according to (9) and (10), we obtain

In view of (f3), we conclude that Thus, we have that

Lemma 4. Then we have that

Proof. For any we have

Then by (5) and Sobolev inequalities we get

Thus, we obtain

Choosing small enough such that since there exists such that

On the other hand, for any it is obvious that Thanks to (f2) and (f3), we obtain that

This fact implies that

is increasing when and decreasing when for almost every Then we have

From above discussions we have that for all Therefore, is bounded below on that is, is well defined.

Let be fixed. By Lemma 3, for each there exist such that By Lemma 3 we have

For our purpose, we just prove that and as

Let

where is defined as (4). By (5) there holds

Hence, is bounded. Let be such that Then there exist and such that

Now, we claim Suppose, by contradiction, that . By for any we have

Thanks to

as It follows a contradiction with equality (13) from two facts: as and is bounded in Hence,

Therefore, we conclude that

Lemma 5.There exists such that for all the infimum is achieved.

Proof. By the definition of there exists a sequence such that Obviously, is bounded in then, up to a subsequence, still denoted by , there exists such that Since the embedding is compact, for all we have

Hence, and

By Lemma 3 we have

Then by Brézis–Lieb lemma and Fatou’s lemma we get

where

That is, one has

for all and all

Now, we claim that In fact, since the situation is analogous, we just prove By contradiction we suppose Hence, let 0 in (14), and we have

Case 1:

If From Lemma (11) we obtain which contradicts our supposition. If by (14) and Lemma 4 we have

The inequality is absurd. Anyway, we have a contradiction.

Case 2:

One the one hand, by Lemma 4 there exists such that

On the other hand, since we obtain Hence, by means of (15) we have

which is a contradiction. Hence, we deduce that

Second, we prove

Since the situation is analogous, we only prove By contradiction we suppose that

Case 1:

According to and Sobolev embedding, we obtain that Let

It is easy to see that small enough and for large enough. Hence, by the continuity of there exists such that

Similarly, there exists such that

Since is compact and . is continuous, there exists such that

Now, we prove that Note that if is small enough, we obtain

for all Hence, there exists such that

That is, any point of with is not the maximizer of . This yields that Similarly, we obtain On the other hand, it is easy to see that

and

for

Then we have

and

for all and all Therefore, according to (14), we conclude for all and all Thus, and Finally, we get that Hence, it follows that is a critical point of . This implies that From (14), (16), and (17) we have

which is a contradiction.

Case 2:

In this case, we can maximize in Indeed, it is possible to show that there exist such that

Hence, there is such that

In the following, we prove that It is noted that and small enough, so we have . Meanwhile, and small enough, then we have

On the other hand, it is obvious that

Hence, we obtain that for all Thus, Hence, That is, is an inner maximizer of in

. Hence, Then it follows from (16) that

which is absurd.

Therefore, from the above arguments we have that

Finally, we prove that is achieved. Since by Lemma 1, there exist such that

Furthermore, it is easy to see that By Lemma 3 we obtain 0 < Since according to Lemma 4, we get

Thanks to (f3), and the norm in is lower semicotinuous, we have

Therefore, and is achieved by

4 Proof of main results

In this section, we prove our main results. First, we prove Theorem 1. In fact, thanks to Lemma 5, we just prove that the minimizer for is indeed a sign-changing solution of problem (1).

Proof of Theorem 1. Since we have

By Lemma 3 and Lemma 4, for we have

If then there exist and such that

Choose

and

In view of (18), it is easy to see that

Let by Lemma 2.3 in [23] there exists a deformation such that

• 

• 

• 

First, we need to prove that

In fact, it follows from Lemma 1 that That is,

On the other hand, we have

which shows that

Therefore, by (b) we have Hence, (19) holds.

In the following, we prove that which contradicts the definition of

Let

and

By the direct calculation we have

Let

By (f3), we have

Then, since we have

Since is C¹ a function and (1,1) is the unique isolated zero point of by using the degree theory we deduce that deg

Hence, combining (19) with (a), we obtain

Consequently, we obtain deg Therefore, for some so that

which contradicts (19).

From the above discussions we deduce that u. is a sign-changing solution for problem (1).

Finally, we prove that . has exactly two nodal domains. To this end, we assume by contradiction that

and

and

Setting we see that Then there exist a unique pair of positive numbers such that Hence,

Moreover, using the fact that we obtain

From Lemma 3(ii) we have that

On the other hand, we have

Hence, by (12) we obtain

which is a contradiction, that is, and has exactly two nodal domains.

By Theorem 1 we obtain a least energy sign-changing solution of problem (1). Next, we prove that the energy of is strictly larger than two times the ground state energy.

Proof of Theorem 2. Similar to the proof of Lemma 5, there exists such that for all and for each , there exists such that By standard arguments (see [10, Cor. 2.13]) the critical points of the functional on are critical points of and we obtain That is, is a ground state solution of (1).

According to Theorem 1, we know that problem (1) has a least energy sign-changing solution , which changes sign only once when

Let

Suppose that As in the proof of Lemma 3, there exist and such that Furthermore, Lemma 3 implies that Therefore, in view of Lemma 3, we have

Hence, it follows that cannot be achieved by a sign-changing function.

References

1. G.F. Carrier, On the nonlinear vibration problem of the elastic string, Q. Appl. Math., 3(2):157– 165, 1945, https://doi.org/10.1090/qam/12351.

2. B. Cheng, J. Chen, B. Zhang, Least energy nodal solutions for Kirchhoff-type Laplacian problems, Math. Methods Appl. Sci., 43(6):3827–3849, 2020, https://doi.org/10.1002/mma.6157.

3. K. Cheng, Q. Gao, Sign-changing solutions for the stationary Kirchhoff problems, involving the fractional Laplacian in R., Acta Math. Sci., Ser. B, Engl. Ed., 38(6):1712–1730, 2018, https://doi.org/10.1016/S0252-9602(18)30841-5.

4. M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., Theory Methods Appl., 30(7):4619–4627, 1997, https://doi.org/10.1016/ S0362-546X(97)00169-7.

5. L. Diening, P. Harjulehto, P. Hästö, M. Ružicka,ˇ Lebesgue and Sobolev spaces with variable exponents, Lect. Notes Math., Vol. 2017, Springer, Berlin, Heidelberg, 2011, https:// doi.org/10.1007/978-3-642-18363-8.

6. X. Fan, D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., 263(2): 424–446, 2001, https://doi.org/10.1006/jmaa.2000.7617.

7. G.M. Figueiredo, N. Ikoma, J.R. Santos Jr., Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213(3):931–979, 2015, https://doi.org/10.1007/s00205-014-0747-8.

8. G.M. Figueiredo, J.R. Santos Jr., Existence of a least energy nodal solution for a Schrödinger– Kirchhoff equation with potential vanishing at infinity, J. Math. Phys., 56(5):051506, 2015, https://doi.org/10.1063/1.4921639.

9. A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., Theory Methods Appl., 94:156–170, 2014, https://doi.org/10.1016/j.na.2013.08.011.

10. X. He, W. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl. (4), 193(2):473–500, 2014, https://doi.org/10.1007/s10231012-0286-6.

11. Z. Liu, Z.-Q. Wang, On Clark’s theorem and its applications to partially sublinear problems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 32(5):1015–1037, 2015, https://doi. org/10.1016/j.anihpc.2014.05.002.

12. A. Mao, Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., Theory Methods Appl., 70(3):1275–1287, 2009, https://doi.org/10.1016/j.na.2008.02.011.

13. C. Miranda, Un’osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital. (2), 3:5–7, 1940.

14. G. Molica Bisci, V.D. Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, Encycl. Math. Appl., Vol. 162, Cambridge Univ. Press, Cambridge, 2016, https://doi.org/10.1017/CBO9781316282397.

15. E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136(5):521–573, 2012, https://doi.org/10.1016/j.bulsci.2011.12.004.

16. V.D. Radulescu, D.D. Repovš,˘ Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis, Monogr. Res. Notes Math., Chapman and Hall/CRC, New York, 2015, https://doi.org/10.1201/b18601.

17. R. Servadei, E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389(2):887–898, 2012, https://doi.org/10.1016/j.jmaa.2011.12.032.

18. W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differ. Equations, 259(4):1256–1274, 2015, https://doi.org/10.1016/j.jde.2015.02.040.

19. X. Tang, B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differ. Equations, 261(4):2384–2402, 2016, https://doi.org/10.1016/j.jde.2016.04.032.

20. D. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys., 61(1):011501, 2020, https://doi.org/10.1063/1.5074163.

21. L. Wang, B. Zhang, Infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents, Appl. Anal., 100(11):2418–2435, 2021, https://doi.org/10.1080/00036811.2019.1688790.

22. T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differ. Equ., 27(4):421–437, 2006, https://doi.org/10.1007/s00526006-0015-3.

23. M. Willem, Minimax Theorems, Prog. Nonlinear Differ. Equ. Appl., Vol. 24, Birkhäuser, Boston, MA, 1996, https://doi.org/10.1007/978-1-4612-4146-1.

24. M. Xiang, D. Hu, B. Zhang, Y. Wang, Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth, J. Math. Anal. Appl., 501(1):124269, 2021, https://doi.org/10.1016/j.jmaa.2020.124269.

25. M. Xiang, B. Zhang, D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal., 178:190–204, 2019, https://doi.org/ 10.1016/j.na.2018.07.016.

26. Z. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317(2):456–463, 2006, https://doi.org/10.1016/ j.jmaa.2005.06.102.

Notas de autor

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