Existence and multiplicity of solutions for fractional p1(x,⋅)&p2(x,⋅)-Laplacian Schrödinger-type equations with Robin boundary conditions
In this paper, we study fractional p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland's variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in R N ∖ Ω ‾ for fractional order p 1 (x , ⋅) & p 2 (x , ⋅) -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results.
Keywords: Schrödinger equations; p1(x,⋅)&p2(x,⋅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p{1}(x,\cdot)\& p{2}(x,\cdot)$\end{document} -Laplacian; Robin boundary conditions; Concave-convex nonlinearities; Krasnoselskii genus theory
Tianqing An, Weichun Bu and Shuai Li contributed equally to this work.
Introduction and the main results
In this paper, we consider fractional -Laplacian Schrödinger-type equations, including concave-convex nonlinearities with nonlocal Robin boundary conditions
1
Graph
where ( , ) is a potential function, ( ) is a bounded domain with the Lipschitz boundary ∂Ω, , , , , are continuous functions, , are positive constants, , are positive weighted functions, , ,
Graph
and
Graph
where . stands for the Cauchy principal value.
Equations (1) arise from general reaction-diffusion equation
2
Graph
where . Problem (2) has applications in biophysics, plasma physics, and chemical reactions. For more details on equation (2), readers are referred to [[1]]. Combining with a -symmetric version of the mountain pass lemma for even functionals and some adequate variational methods, Mihăilescu [[3]] proved that the equations
3
Graph
have infinitely many weak solutions. In addition, Chung and Toan [[4]] considered a class of fractional Laplacian problems
4
Graph
using variational techniques and Ekeland's variational principle. The authors used the variational techniques to discuss the results of the existence of solutions in fractional cases [[5]–[7]]. In addition, Heidarkhani et al. [[8]–[10]] studied the existence results of variable exponent equations using variational methods and established the critical point theory. Zuo et al. [[11]] investigated the existence and multiplicity of solutions for the fractional Choquard problems with variable order. On a similar issue, a related study was conducted by Biswas et al. For more details, see [[12]].
The classical Schrödinger equation is of the following form:
Graph
where V, φ denote the potential function and wave function, respectively, and i, h are constants ([[13]]). Recently, Xiang et al. in [[14]] and Bu et al. in [[15]] discussed the fractional Laplace operator Schrödinger equations with variable order and Schrödinger–Kirchhof-type equations, respectively.
The critical local problem involving concave-convex nonlinearities was first studied by Ambrosetti et al. in [[16]]. Subsequently, variational methods were used [[17]] to discuss the following equations:
5
Graph
with the variable order concave-convex term. For other similar types of equations, see [[18]] and the references therein.
The Robin and Neumann boundary problems are interesting topics [[20]]. Mugnai et al. [[21]] investigated fractional p-Laplacian problems with nonlocal Neumann boundary conditions. Moreover, Deng [[22]] considered the following equations:
6
Graph
For double-phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian, Manouni et al. [[23]] proved the existence of solutions by variational tools, truncation techniques, and comparison methods. In many papers, the Robin and Neumann boundary problems of fractional equations were studied in different ways; e.g., the Morse theory was used in [[24]], the mountain pass lemma in [[26]], Ekeland's variational principle in [[28]], and the topological degree in [[30]].
To our knowledge, there is no previous work on the problem (1). This paper is devoted to this topic. We obtain new results by applying the mountain pass lemma, Ekeland's variational principle, the fountain theorem, and the Krasnoselskii genus theory. Our problem differs from problems (3), (4), and (5) in that we discuss Robin boundary conditions, and it also differs from problem the (6) in that we consider -Laplacian Schrödinger-type equations with concave-convex nonlinearities.
Before stating the main results, we introduce the basic assumptions.
-
is a symmetric and continuous function, that is,
-
Graph
• with
-
Graph
• and
Graph
- such that . Let , the fractional critical exponent be defined as and for all .
-
.
-
is a continuous function, satisfying , for all , .
-
and are weighted functions in and satisfy such that and for all , such that and for all . Here, and are conjugate exponents of the functions and , respectively.
The main results of this paper are as follows:
Theorem 1.1
Assume that assumptions (P), (G), (V), and (H) hold. Equations (1) have two nontrivial weak solutions.
Theorem 1.2
Assume that assumptions (P), (G), (V), and (H) hold. Then, equations (1) have infinitely many nontrivial weak solutions inX.
Theorem 1.3
Assume that assumptions (P), (G), (V), and (H) hold. Then, equations (1) possess infinitely many solutions.
In Sect. 2, we state some basic results of the Lebesgue space . In Sect. 3, we introduce the workspaces associated with equations (1). In Sect. 4, we verify the conditions and prove Theorem 1.1 by the mountain pass lemma and Ekeland's variational principle. In Sect. 5, we prove Theorem 1.2 by applying the fountain theorem. Finally, using the Krasnoselskii genus theory, we give the proof of Theorem 1.3.
Preliminaries
In this section, we recall some basic results of the Lebesgue space with a variable exponent. Assume that domain Ω is bounded in with the Lipschitz boundary ∂Ω. Let
Graph
where .
The variable exponent Lebesgue space , which is defined by
Graph
equipped with the Luxemburg norm
Graph
where is a separable, uniformly convex, and reflexive Banach space [[31]].
Let be the conjugate space of and ( and are conjugate indices to each other). For and , the Hölder inequality
7
Graph
holds. If ( ) and
Graph
for all , there exists
Graph
Lemma 2.1
([[32]]) Let be the modular of the space, and defined by . Then, the following properties hold:
-
;
-
;
-
.
Lemma 2.2
([[32]]) If with , then
-
;
-
;
-
a. e. in Ω and .
Lemma 2.3
([[33]]) Let , be measurable functions such that and , for any . Then, there is
Graph
with
, .
The basic properties of functionals and operators
In this section, we state some properties of functionals and operators, and give the definition of weak solutions of equations (1) with Robin boundary conditions. We first introduce the workspaces and associated with equations (1).
The fractional variable Sobolev space is given by
Graph
Set
Graph
as the variable exponent Gagliardo seminorm. W is a Banach space with the norm
Graph
We take into account three continuous functions and . From condition (P), we know that
8
Graph
Lemma 3.1
([[34]]) Suppose that is a bounded open domain and (8) holds. Then, Wis a separable and reflexive space.
Lemma 3.2
([[35]]) Let smooth bounded domain , for with , and for . Suppose that continuous function satisfies
Graph
There exists a positive constant
such that for every
, it holds that
Graph
Then, the embedding for all is compact.
Lemma 3.3
([[35]]) If and
Graph
There exists a positive constant
such that
Graph
Then, the embedding is compact.
Define nonlinear map
9
Graph
for all , has the following properties.
Lemma 3.4
([[28]])
-
is a bounded and strictly monotone operator;
-
is a mapping of
, i.e., if inWand , then inW;
-
is a homeomorphism.
Define function
Graph
which is related to (9). The derivative of S is
Graph
for all ; for more details, see [[34]].
Let
Graph
equipped with the norm
Graph
where
Graph
and
Graph
with .
Lemma 3.5
([[28]]) Assume that assumptions (P), (G), and (V) hold. Then, is a reflexive Banach space.
The norm on is equivalent to
Graph
where the modular is defined by
10
Graph
Lemma 3.6
Assume that assumptions (P), (G), and (V) hold. The following properties hold:
-
;
-
;
-
;
-
.
Let with norm , which is a separable and reflexive Banach space. The dual space of X is . The modular . We have the following result.
Lemma 3.7
([[28]]) Assume that assumptions (P), (G), and (V) hold. Then, from (10), the following properties hold:
-
The function
is of class
;
-
The strictly monotone operator
is coercive, then
-
Graph
-
is a mapping of type
, that is, if inXand , then inX.
Lemma 3.8
([[35]]) Assume that assumptions (P), (G), (V), and (H) hold. Then, for any with for all , there is a positive constant such that
Graph
Moreover, this embedding is compact.
Lemma 3.9
([[35]]) Assume that assumptions (P), (G), (V), and (H) hold. Then, for any with for all , there is a positive constant such that
Graph
Moreover, this embedding is compact.
More precisely, we now present the divergence theorem and the analogous formula for the partition integral formula in nonlocal case [[37]].
Lemma 3.10
([[29]]) Let the hypotheses (P) hold, and letφbe any bounded -function in . Then,
Graph
Lemma 3.11
Let the hypotheses (P) hold. Suppose thatφandvare bounded -functions in . Then,
Graph
for every
.
Proof
According to symmetry, we obtain
Graph
□
Lemma 3.12
Assuming that assumption (P) holds and lettingφbe a weak solutions of equations (1), we have
Graph
Lemma 3.13
Assuming that assumptions (P), (G), (V), and (H) hold, let be a energy function defined by
Graph
for every
. Then, any critical point of is a weak solution of equations (1).
Proof of Theorem 1.1
To prove Theorem 1.1, we need a well-known mountain pass lemma.
Theorem 4.1
Let
X
be a real Banach space and
with
. Assume that the following conditions hold:
-
satisfies
conditions;
-
there exist
such that
, for all , with ;
-
there exists
, satisfying such that .
Then, has a critical value , that is,
Graph
where
.
Definition 1
Let X be a Banach space, . We say that satisfies the conditions if every sequence satisfying
Graph
has a convergent subsequence in X.
Next, we prove that the defined in Lemma 3.13 satisfies the conditions.
Lemma 4.1
Assume that assumptions (P), (G), (V), and (H) hold. Then, the sequence is bounded inX.
Proof
According to (H), we get
Graph
so from Lemmas 3.8 and 3.9, there exist constants and such that
11
Graph
Let and
Graph
Thus, by the Hölder inequality and Lemma 2.3, for all with , we obtain
12
Graph
and
13
Graph
We use the counterfactual method. Suppose , . Combining conditions (P), (G), (V), (H), and Lemma 3.8 and letting , where , , , we have
Graph
In addition, we obtain because , . Due to
Graph
there is a contradiction. Thus, is bounded. □
Inspired by [[15]], we have the following lemma.
Lemma 4.2
Assume that assumptions (P), (G), (V), and (H) hold. Then, satisfies the conditions.
Proof
According to Lemma 4.1, is bounded, that is, there is a subsequence and in X such that
Graph
Due to in , then . We get
Graph
Since and in X are sequentially weakly lower semi-continuous, for and measurable for all , we obtain
Graph
Hence, is uniformly integrable in . Then, using the Vitali convergence theorem, we get
Graph
Similarly, there is
Graph
We need to prove that is strongly convergent,
Graph
A discussion similar to Lemma 3.7 gives that in X. Combining the Definition 1 and the Lemma 4.1, we complete the proof. □
Lemma 4.3
Assume that assumptions (P), (G), (V), and (H) hold. There exist and such that, for all with ,
Graph
holds.
Proof
Combining (12) with (13), for any with , we have
Graph
Let
Graph
where . Then, there exists such that . Choosing , we get
Graph
for . □
Lemma 4.4
Assume that assumptions (P), (G), (V), and (H) hold. Then, there existsυ, which satisfies . Then, there exists such that
Graph
Proof
Choosing such that , and for small enough, we obtain
Graph
with the fact that . Thus, with . The proof is proved by letting . □
Proof of Theorem 1.1
Combining Lemmas 4.1 and 4.2, it can be inferred that satisfies conditions. According to Lemmas 4.3 and 4.4, we know that satisfies the mountain pass lemma. Therefore, we have a subsequence and such that in X by Lemma 4.1 and . Therefore, , that is, is a solution of problem (1) with positive energy.
Next, we will apply Ekeland's variational principle to prove that (1) has a solution with negative energy.
By Lemma 4.3, we derive that
Graph
where ρ is the positive constant introduced in Lemma 4.3.
From condition (H), there exist and an open set such that
Graph
and we get
Graph
Hence,
14
Graph
By Lemma 2.1 and , we conclude
Graph
For sufficiently small , let such that , , for all and in Ω. Then, by applying (14), it follows that
Graph
Since , we have .
In addition, combining the Hölder inequality and inequality (11), for any , we have
Graph
This fact gives
15
Graph
Set
Graph
By (15), is lower bounded on and . Using Ekeland's variational principle, there exists such that
16
Graph
Since
Graph
we have . Define function by
Graph
which implies from (16). Then, is a minimum point of ζ, and we have
Graph
for small and any . Hence,
Graph
Let , then . Replace v with −v. Then, we obtain . Thus, . We infer that there exists a sequence such that
Graph
By Lemma 4.2, there is in X. Then, we have and , that is, is another solution of equations (1) with negative energy, which ends the proof. □
Here, we give an example of application of Theorem 1.1.
Example 4.1
Let . Consider the problem
17
Graph
By simple calculations, we obtain , , , , . Conditions (P), (G), (H), and (V) are satisfied. We observe that all assumptions of Theorem 1.1 are fulfilled. Hence, Theorem 1.1 implies that problem (17) admits two nontrivial weak solutions.
Proof of Theorem 1.2
To prove Theorem 1.2, we first recall the following lemmas.
Lemma 5.1
([[15]]) LetXbe a reflexive and separable Banach space. Then, there are and such that
Graph
and
Graph
Denote
Graph
Lemma 5.2
([[15]]) Assume that , , for any and denote
Graph
then
.
Now, we recall the fountain theorem.
Theorem 5.1
([[15]]) LetXbe a real Banach space and be a even functional satisfying the conditions. There exists such that for every . Then, the following conditions hold:
-
;
-
as
.
Then, possesses a series of critical points such that .
Lemma 5.3
Assume that assumptions (P), (G), (V), and (H) hold. There exists such that
Graph
Proof
Let . For and , there exists φ̂ such that
Graph
with . Taking , for sufficiently small t, it follows that
Graph
□
Lemma 5.4
Assume that assumptions (P), (G), (V), and (H) hold. There exists such that
Graph
Proof
According to Lemma 5.2, for , we obtain
Graph
Let
Graph
and there exists a constant C̃ such that , where . Therefore,
Graph
Choose
Graph
Since , we have as . By the choice of with such that , we obtain
Graph
□
Proof of Theorem 1.2
Let hypotheses (P), (G), (V), and (H) be satisfied. By Lemma 4.2, satisfies conditions. Under the definition of in Lemma 3.13, it follows that and is an even function. Therefore, from Lemmas 5.3 and 5.4, it can be deduced that satisfies Theorem 5.1. Then, possesses a series of critical points as . In conclusion, equations (1) possess infinitely many nontrivial weak solutions. □
Here, we give an example of application of Theorem 1.2.
Example 5.1
Let . Consider the problem
18
Graph
By simple calculations, we obtain , , , , , , , , and . That is, conditions (P), (G), (H), and (V) are satisfied. We observe that all assumptions of Theorem 1.2 are fulfilled. Hence, Theorem 1.2 implies that problem (18) admits infinitely many nontrivial weak solutions.
Proof of Theorem 1.3
We give some results with the aid of the Krasnoselskii genus. Let E be a real Banach space and set
Graph
Let and . We define genus
Graph
If the mapping ψ does not exist for any , and set . If is a subset consisting of a finite number of pairs of points, then, . Furthermore, from definition, .
Lemma 6.1
([[18]]) Let and∂Ω be the boundary of an open, symmetric, and bounded subset with . Then, .
Corollary 6.1
([[18]]) .
Theorem 6.1
([[18]]) Let be a functional satisfying the conditions and assume that
-
is bounded from below and even;
-
there is a compact set
such that
and
.
Then, has at leastkpairs of distinct critical points whose corresponding critical values are all less than .
Proof of Theorem 1.3
Combining (12), (13), and the Hölder inequality (7) for , we obtain
Graph
Since , for large enough, is bounded from below. is an even function by the definition and . Moreover, is coercive in X and satisfies the conditions by Lemma 4.2. Let
Graph
We obtain
Graph
Now we prove that for any , there is . For each k, we take k disjoint open sets such that . For , let with , and
Graph
Since each norm on is equivalent, there is such that with , which means that . Set
Graph
Combining the compactness of and for all ,
Graph
Sine , there exist and such that
Graph
Thus, for all . Furthermore, such that for all k, and the assertion is proved. Each is a critical value by the Krasnoselskii genus theory. Combining Theorem 6.1, has at least k pairs of different critical points. In addition, since k is arbitrary, we obtain an infinite number of critical points of equations (1). □
Acknowledgements
This work is supported by the Postgraduate Research Practice Innovation Program of Jiangsu Province under Grant (KYCX23-0669); and the Doctoral Foundation of Fuyang Normal University (2023KYQD0044).
Author contributions
Zhenfeng Zhang wrote the main manuscript text and Tianqing An, Weichun Bu, and Shuai Li verified this article. All authors reviewed the manuscript.
Funding
This work is supported by the Postgraduate Research Practice Innovation Program of Jiangsu Province under Grant (KYCX23-0669); and the Doctoral Foundation of Fuyang Normal University (2023KYQD0044).
Data Availability
No datasets were generated or analysed during the current study.
Declarations
Competing interests
The authors declare no competing interests.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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]
By Zhenfeng Zhang; Tianqing An; Weichun Bu and Shuai Li
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