Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
-
Shuaishuai Liang
Abstract
In this article, we consider the following double critical fractional Schrödinger-Poisson system involving p-Laplacian in
where
1 Introduction
In this article, we are concerned with the multiplicity and concentration of positive solutions for the following fractional p-Laplacian Schrödinger-Poisson system involving double critical nonlinearities in
where
where
Benci and Fortunato [8] proposed the following famous Schrödinger-Poisson system:
the system 1.2 as the model to describe the solitary waves for the nonlinear stationary Schrödinger equations interacting with the electrostatic field which is not a priori assigned. Here,
In recent years, many scholars focused on the study of the Schrödinger-Poisson system. By using the monotonicity method introduced by Jeanjean [25] and a truncation argument, Azzollini et al. [7] considered a nontrivial positive radial solution for the following Schrödinger-Poisson system:
which was the first contribution to the existence of solutions for the Schrödinger-Poisson system. In [47], Ruiz considered the classical Schrödinger-Poisson system
Applying the variational method, Ruiz obtained the existence of the solution to the above problem in both cases
Once we turn our attention on the fractional setting, there has some interesting results about the Schrödinger-Possion system. For example, Zhang et al. [55] explored the following fractional nonlinear Schrödinger-Poisson system:
Under the Berestycki-Lions conditions (see Zhang et al. [55] (H2)–(H4)) combined with the perturbation approach, they obtained the existence of positive solutions and the asymptotics of solutions for system (1.4) as the nonlinearity
where
On the other hand, to our best knowledge, most of the scholars consider the existence of solutions for Schrödinger-Possion with the nonlinearity
By the well-known Ljusternik-Schnirelmann category theory and the method of generalized Nehari mainfold from Szulkin [49], they obtain the concentration of solutions for the fractional Schrödinger-Poisson system. Meng and He [40] were concerned with the following Schrödinger-Poisson system:
With help of the truncation technique and the genus theory, they obtained the existence and multiplicity of normalized solutions for system (1.5). Feng [20] considered the critical Schrödinger-Poisson-type system of the form:
where the partial potential
where
where
Inspired by the aforementioned works, we consider concentrating solutions for the double critical fractional Schrödinger-Poisson system with p-laplacian in
There is an open bounded domain
And the nonlinearity
There exist
There exist
The function
There exists
Now, we are ready to state our results in this article.
Theorem 1.1
Assume that (
Remark 1.1
There is no doubt that we shall encounter the following difficulties when we deal with system (1.1):
The nonlinearity satisfying critical growth in system (1.1) means that there does not exist the convergence of bounded (PS) sequences. It is more complicated to obtain the critical value of the mountain pass, due to the appearance of double critical terms in system (1.1), which is the major novelty in this article. We shall use the concentration-compactness principle to overcome this obstacle. In addition, there are two nonlocal critical convolution terms in the Schrödinger-Poisson system, and it is natural to consider how the interaction between the nonlocal term and the nonlinear term will affect the existence of system (1.1).
The nonlinearity
Remark 1.2
In this article, our work considers the new Poisson term involving the upper exponent
This article is organized as follows: In Section 2, we introduce the variational setting and give preliminary lemmas. In Section 3, we will consider the modified problem by using truncated function. In Section 4, we study the autonomous problem. In Section 5, we are devoted to the proof of Theorem 1.1.
2 Preliminaries
In this section, first, we introduce some notations
Next, we define the fractional Sobolev space
and the
We also know that the fractional Sobolev space
which is equivalent to the following norm:
Especially p = 2, we have the following:
Now we introduce the workspace framework as the follow,
and
Lemma 2.1
(see Adams [1]) Let
so that
Moreover,
Lemma 2.2
(see Lions [31]) Let
where
From the Lax-Milgram theorem, for each
For the weak solution
where
To deal with the difficulties caused by the critical Poisson term, we need to use the following well-known Hardy-Littlewood-Sobolev inequality.
Proposition 2.1
(see Lieb and Loss [29]) Let
Now, we consider that if
has well defined and we need that
where
Now, we use
and we have the following relationship for the constant
Applying the change of variable
where
Next, we introduce the following results which are similar to Du et al. [15]. For reader’s benefit, here we give a detailed proof.
Lemma 2.3
For any
for each
and
where
If
If
in
Proof
and
That is,
Let
Note that
Thus, from (2.6) and (2.7), we know that conclusion
So,
3 The modified problem
In this section, we mainly investigate the existence of positive solutions to system (1.1). We use the change of variable
Now, we consider the modified equation, there exist
and
where
(i)
(ii)
for each
It is clear that equation (3.1) and the following equation (3.4) are equivalent,
where
where
for any
Next, we suppose that
Lemma 3.1
The energy functional
there exist
there exists
Proof
According to
and
By (3.5) and Lemma 2.3-(ii), we have
Since
(ii) We fix the function
Taking
We assume that
and
where
Lemma 3.2
Suppose that
For each
There exists
The map
If there is a sequence
Proof
and
Thus, according to the definition of
The above inequality is impossible, so we prove
Thus, there exists
Fix
Combining with (3.13) and
Combining
for all
Thus, we can obtain that
According to the definition of
Thus,
Let us consider the maps
by
Proposition 3.1
Suppose that hypotheses
where
Assume that
u is a critical point of
Lemma 3.3
Let
Proof
Suppose that
Then, according to
Thus, we deduce that
and we denote
This completes the proof.□
In order to prove that
Lemma 3.4
Let
weakly in the sense of a measure where
Then, there is a countable sequence of points
and
where
and
Moreover, if
Proof
Since
In order to prove the possible concentration at finite points, we first show that
where the function
Since
Since the Riesz potential defines a linear operator and
So, we have
where
In Lions [30], we note that
where
and so we obtain
Then, for
Using the above formula and
Combining this and (3.25), we have
So, (3.24) is demonstrated.
Now, for any
From (3.24), we have
Passing to the limit as
Applying Lemma 1.2 in Lions [30], we know (3.20) holds. Besides, let
By the definition of
By (3.24) and
Passing to the limit as
Applying Lemma 1.2 in Lions [30], we know (3.22) holds. Let
This completes the proof of (3.21).
Then, we are going to demonstrate the possible loss of mass at infinity. For
as
Using the Hardy-Littlewood-Sobolev inequality, we infer that
which means
According to the definition of
which means
Besides, if
Thus, we can deduce that
And so, for each open set
It follows that
Next, we give the proof of the compactness result.
Proposition 3.2
Proof
Let
For proving the Proposition 3.2, the proof will be divided into the following three steps.
Step I. We claim that
In fact, define
Due to the boundedness of
Then, by (3.28), the definition of
and
By the absolute continuity of the Lebesgue integral and the definition of
Summing up, from (3.29) to (3.33), taking the limit as
So, we have
Step II. We claim that
In fact, from Lemmas 3.3 and 3.4, let
Analogously, by the boundedness of
By the definition of
Thus, given that Lemma 3.4, we obtain
Similarly, we obtain that
and
Summing up, from (3.34)–(3.38), similar to Step II, taking the limit as
Step III. We claim that
In fact, suppose by contradiction that there exists
and
From (3.39) and (3.40), we know that
If
This fact implies that
If
From this fact, one has
By using the selection of
This obviously contradicts with
On the other hand, if
and
By (3.43) and (3.44), we obtain
If
This implies that
If
This means that
Moreover, we have
Similarly, this contradicts the selection of
and
So, we have
Now, from (3.49) to (3.52), we have
This completes the proof of Proposition 3.2.□
According to Propositions 3.1 and 3.2, we obtain the following result.
Corollary 3.1
4 The autonomous problem
In this section, we mainly consider the following equation (4.1):
According to equation (4.1), we give the energy functional is
which is well defined on the Sobolev space
and norm in this space is given
From
for any
and
where
Next, we establish the following mountain pass geometry.
Lemma 4.1
One of the following holds:
there exist
there exist
Similar to the results of Lemma 3.1 and Proposition 3.1 in Section 3, we introduce the following lemma.
Lemma 4.2
Assume that
For each
There exists
The map
If there is a sequence
Now, we consider the maps
where
Proposition 4.1
Suppose that the hypotheses
where
Assume that
u is a critical point of
Remark 4.1
As in Szulkin and Weth [50], we can know that the following equalities hold:
where
Lemma 4.3
There exists
Proof
Let
By
which means that
Combining with (4.2), we can know that
Then, as
Now, we use the following lemma to verify the weak limit of the
Lemma 4.4
Assume that
if
if
Proof
According to the above arguments, we can obtain
Thus, we know that
we assume that if there exists a contradiction, such that
According to the boundedness of
Then, from (4.4) and taking
We assume that
Combining (4.5) and Lemma 2.1, we can obtain
Letting
Meanwhile,
Lemma 4.5
Equation (3.1) has a positive ground-state solution.
Proof
Let
Now let
Lemma 4.6
The proof of this part will be given in the next chapter and will be omitted here.
5 Multiplicity of solution to equation (3.1)
In this section, we will consider the multiplicity of solutions to equation (3.1). First, we introduce the following lemma.
Lemma 5.1
Assume that
Proof
According to
If we assume that (5.1) is false. Thus, for all
In
So,
Next, we consider the boundedness of
as
According to Lemma 4 of Thin [52], for any
as
Combining (3.49), (3.50), (3.51), (5.3), and (5.4), we can obtain
Via Fatou lemma, (5.2), and condition
which is a contradiction, so we prove the boundness of
Obviously, this is impossible, hence we know
Next, for proving
which is impossible. So,
Next, we will combine the topology of the set
We choose
where
There is a unique
We denote that
where
Proceeding line by line as in Qu and He [44] Lemma 5.2, we can obtain the following result:
Lemma 5.2
It follows that
Next, we define the barycenter map, for any
Define
In the following, we define the map
Lemma 5.3
We have the following limit:
Proof
We assume that it is false, then there exist
Combining with
It is easy to obtain
which contradicts relation (5.5).□
Let
and
For each
Lemma 5.4
For any
Proof
Let
Thus, it is sufficient to prove a sequence
In fact, from
According to Lemma 5.1, there exists
Using the
Therefore, the proof is completed.□
Now, we give the multiplicity result for equation (3.4).
Theorem 5.1
Suppose that
Proof
For any
Set
where
Moreover, there is
is well defined for any
Therefore, for
Next, we will consider equation (3.4) via a variant of the Moser iteration argument in Moser [39].
Lemma 5.5
Let
Then, up to a subsequence,
Proof
We take
Then,
According to
Due to the function
By the Sobolev inequality and (5.11), we infer that
and
Via the definition of
where
Next, we estimate the first term on the right side of (5.14), by using the Hardy-Littlewood-Sobolev inequality, we can obtain
There exists
for any
Due to
Combining (5.16) and (5.17), we have
Next, we calculate another term in (5.14)
We take
Combining (5.16) and (5.17), we have
Together with (5.14)–(5.16), (5.18)–(5.20), and (5.21), we deduce that
Via the famous Fatou’s lemma and we take
That means
Letting
Note that
Substituting (5.25) into (5.26) and repeating the process, we have
We can easily obtain
Using the D’Alembert principle, we have
which means that
Therefore, equation (5.9) can be rewritten in the following form:
where
where
there exists
Therefore, arguing as in the proof of Lemma 2.6 in Alves and Miyagaki [2], we have that
uniformly in
Proof of Theorem 1.1
We suppose that there exist some sequences
Assume that there exist some sequences
From the above fact
We know that
From Lemma 5.1, there exists
So, we deduce that for any
Therefore, we reach a contradiction with
According to
And up to a subsequence, we can suppose that
We can easily prove that this is impossible. So (5.30) holds. Taking into account (5.30) and Theorem 5.1, we can deduce that the maximum points
Therefore, the proof of Theorem 1.1 is completed.□
-
Funding information: Shaoyun Shi was supported by the NSFC (Grant No. 12271203). Shuaishuai Liang acknowledges the financial support provided by the China Scholarship Council (No. 202306170138). Yueqiang Song was supported by the Science and Technology Development Plan Project of Jilin Province, China (No. 20230101287JC), the National Natural Science Foundation of China (No. 12001061), and the Young outstanding talents project of Scientific Innovation and entrepreneurship in Jilin (No. 20240601048RC).
-
Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
-
Conflict of interest: The authors state no conflict of interest.
-
Competing interests: The authors declare that they have no competing interests.
-
Data availability statement: Date sharing is not applicable to this article as no new data were created or analyzed in this study.
References
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. Search in Google Scholar
[2] C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in RN via penalization method, Calc. Var. Partial Differential Equations 55 (2016), 47, https://doi.org/10.1007/s00526-016-0983-x. Search in Google Scholar
[3] A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math. 76 (2008), 257–274, https://doi.org/10.1007/s00032-008-0094-z. Search in Google Scholar
[4] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on RN, Birkhäuser, Verlag, 2006, https://doi.org/10.1007/3-7643-7396-2. Search in Google Scholar
[5] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math. 10 (2008), 391–404, https://doi.org/10.1142/S021919970800282X. Search in Google Scholar
[6] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl. 345 (2008), 90–108, https://doi.org/10.1016/j.jmaa.2008.03.057. Search in Google Scholar
[7] A. Azzollini, P. d’Avenia, and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. Henri Poincaré, Anal. Non Linéire. 27 (2010), 791–799, https://doi.org/10.1016/j.anihpc.2009.11.012. Search in Google Scholar
[8] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Top. Meth. Nonlinear Anal. 11 (1998), 283–293, https://doi.org/10.12775/TMNA.1998.019. Search in Google Scholar
[9] M. Bhakta, K. Perera, and F. Sk, A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities, Adv. Nonlinear Stud. 23 (2023), 20230103, https://doi.org/10.1515/ans-2023-0103. Search in Google Scholar
[10] D. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2, Comm. Partial Differential Equations 17 (1992), 407–435, https://doi.org/10.1080/03605309208820848. Search in Google Scholar
[11] S. Chen, V. D. Rădulescu, and X. Tang, Multiple normalized solutions for the planar Schrödinger-Poisson system with critical exponential growth, Math. Z. 306 (2024), no. 3, Paper No. 50, 32 pp. https://doi.org/10.1007/s00209-024-03432-9. Search in Google Scholar
[12] G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal. 7 (2003), 417–423. Search in Google Scholar
[13] J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differential Equations 3 (1995), 493–512, https://doi.org/10.1007/BF01187898. Search in Google Scholar
[14] M. Del Pino and P. L. Felmer, Local Mountain Pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137, https://doi.org/10.1007/BF01189950. Search in Google Scholar
[15] Y. Du, J. B. Su, and C. Wang, On a quasilinear Schrödinger-Poisson system, J. Math. Anal. Appl. 1 (2022), 125446, https://doi.org/10.1016/j.jmaa.2021.125446. Search in Google Scholar
[16] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353, https://doi.org/10.1016/0022-247X(74)90025-0. Search in Google Scholar
[17] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573, https://doi.org/10.1016/j.bulsci.2011.12.004. Search in Google Scholar
[18] P. D’Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud. 2 (2002), 177–192, https://doi.org/10.1515/ans-2002-0205. Search in Google Scholar
[19] P. Felmer, A. Quaas, and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A. 142 (2012), 1237–1262, https://doi.org/10.1017/S0308210511000746. Search in Google Scholar
[20] X. J. Feng, Ground state solution for a class of Schrödinger-Poisson-type systems with partial potential, Z. Angew. Math. Phys. 71 (2020), 1–16, https://doi.org/10.1007/s00033-020-1254-4. Search in Google Scholar
[21] X. He, Positive solutions for fractional Schrödinger-Poisson system with doubly critical exponents, Appl. Math. Lett. 120 (2021), 107190. Search in Google Scholar
[22] X. He and D. B. Wang, Multiple bound state solutions for fractional Schrödinger-Poisson systems with critical nonlocal terms, J. Geom. Anal. 33 (2023), 194, https://doi.org/10.1007/s12220-023-01247-4. Search in Google Scholar
[23] X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations 55 (2016), 91, https://doi.org/10.1007/s00526-016-1045-0. Search in Google Scholar
[24] X. He, X. Zhao, and W. Zou, The Benci-Cerami problem for the fractional Choquard equation with critical exponent, Manuscripta Math. 170 (2023), 193–242, https://doi.org/10.1007/s00229-021-01362-y. Search in Google Scholar
[25] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on RN, Proc. R. Soc. Edinb. Sect. A. 129 (1999), 787–809, https://doi.org/10.1017/S0308210500013147. Search in Google Scholar
[26] K. Jin and L. Wang, Infinitely many solutions for the nonlinear Schrödinger-Poisson system, J. Dyn. Control Syst. 29 (2023), 1299–1322, https://doi.org/10.1007/s10883-022-09636-8. Search in Google Scholar
[27] S. Liang and V. D. Rădulescu, Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity, Anal. Math. Phys. 10 (2020), no. 45, 1–31, https://doi.org/10.1007/s13324-020-00386-z. Search in Google Scholar
[28] C. Lei, V. D. Rădulescu, and B. L. Zhang, Ground states of the Schrödinger-Poisson-Slater equation with critical growth, RACSAM Rev. R. Acad. A. 117 (2023), 128, https://doi.org/10.1007/s13398-023-01457-z. Search in Google Scholar
[29] E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, American Mathematical Society, vol. 14. Providence, 2001.Search in Google Scholar
[30] P. L. Lions, The concentration compactness principle in the calculus of variations, The locally compact case, I, II, Ann. Inst. Poincaré Anal. Nonlin. 1 (1984), 109–145, 223–283, https://doi.org/10.1016/S0294-1449(16)30428-0. Search in Google Scholar
[31] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063–1072, https://doi.org/10.1016/0362-546X(80)90016-4. Search in Google Scholar
[32] X. Liang and B. L. Zhang, A modified zero energy critical point theory with applications to several nonlocal problems, Electron. J. Qual. Theory Differ. Equ. 6 (2024), 20, https://doi.org/10.14232/ejqtde.2024.1.6. Search in Google Scholar
[33] Y. Li, B. L. Zhang, and X. Han, Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations, Adv. Nonlinear Anal. 12 (2023), 20220293, https://doi.org/10.1515/anona-2022-0293. Search in Google Scholar
[34] Y. Li, V. D. Rădulescu, and B. Zhang, Critical planar Schrödinger-Poisson equations: existence, multiplicity and concentration, Math. Z. 307 (2024), 43, https://doi.org/10.1007/s00209-024-03520-w. Search in Google Scholar
[35] G. B. Li and S. S. Yan, Eigenvalue problems for quasilinear elliptic equations on RN, Comm. Partial Differential Equations 14 (1989), 1291–1314, https://doi.org/10.1080/03605308908820654. Search in Google Scholar
[36] Z. Liu, Z. Zhang, and S. Huang, Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation, J. Differential Equations 266 (2019), 5912–5941, https://doi.org/10.1016/j.jde.2018.10.048. Search in Google Scholar
[37] Z. Liu, L. Tao, D. Zhang, S. Liang, and Y. Song, Critical nonlocal Schrödinger-Poisson system on the Heisenberg group, Adv. Nonlinear Anal. 11 (2022), 482–502, https://doi.org/10.1515/anona-2021-0203. Search in Google Scholar
[38] S. L. Liu and H. B. Chen, Existence of ground-state solutions for p-Choquard equations with singular potential and doubly critical exponents, Math. Nachr. 296 (2023), 2467–2502, https://doi.org/10.1002/mana.202100255. Search in Google Scholar
[39] J. Moser, A new proof of De Giorgias theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468, https://doi.org/10.1002/cpa.3160130308. Search in Google Scholar
[40] Y. X. Meng and X. M. He, Normalized solutions for the Schrödinger-Poisson system with doubly critical growth, Topol. Methods Nonlinear Anal. 62 (2023), 509–534, https://doi.org/10.12775/TMNA.2022.075. Search in Google Scholar
[41] N. S. Papageorgiou, J. Zhang, and W. Zhang, Global existence and multiplicity of solutions for nonlinear singular eigenvalue problems, Discrete Contin. Dyn. Syst. Ser. S (2024), https://doi.org/10.3934/dcdss.2024018. Search in Google Scholar
[42] P. Pucci, M. Xiang, and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN, Calc. Var. Partial Differential Equations 54 (2015), 2785–2806, https://doi.org/10.1007/s00526-015-0883-5. Search in Google Scholar
[43] P. Pucci, L. Wang, and B. Zhang, Bifurcation and regularity of entire solutions for the planar nonlinear Schrödinger-Poisson system, Math. Ann. 389 (2024), 4265–4300, https://doi.org/10.1007/s00208-023-02752-1. Search in Google Scholar
[44] S. Qu and X. He, On the number of concentrating solutions of a fractional Schrödinger-Poisson system with doubly critical growth, Anal. Math. Phys. 12 (2022), 59, https://doi.org/10.1007/s13324-022-00675-9. Search in Google Scholar
[45] D. Qin, V. D. Rădulescu, and X. Tang, Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations, J. Differential Equations 275 (2021), 652–683, https://doi.org/10.1016/j.jde.2020.11.021. Search in Google Scholar
[46] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z angew Math Phys. 43 (1992), 270–290, https://doi.org/10.1006/jdeq.2000.3946. Search in Google Scholar
[47] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2006), 655–674, https://doi.org/10.1016/j.jfa.2006.04.005. Search in Google Scholar
[48] Y. Su, Positive solution to Schrödinger equation with singular potential and double critical exponents, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31 (2020), 667–698, https://doi.org/10.4171/RLM/910. Search in Google Scholar
[49] A. Szulkin, Ljusternik-Schnirelmann theory on C1-manifolds, Ann. Inst. H. Poincaré Anal. NonLinéaire 5 (1988), 119–139, https://doi.org/10.1016/S0294-1449(16)30348-1. Search in Google Scholar
[50] A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, International Press, Boston, 2010, pp. 597–632. Search in Google Scholar
[51] K. Teng, Existence of ground-state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations 261 (2016), 3061–3106, https://doi.org/10.1016/j.jde.2016.05.022. Search in Google Scholar
[52] N. V. Thin, Multiplicity and concentration of solutions to a fractional p-Laplace problem with exponential growth, Ann. Fenn. Math. 47 (2022), 603–639, https://doi.org/10.54330/afm.115564. Search in Google Scholar
[53] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math. 20 (1967), 721–747, https://doi.org/10.1002/cpa.3160200406. Search in Google Scholar
[54] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. Boston, MA, 1996. Search in Google Scholar
[55] J. Zhang, J. M. do Ó, and M. Squassina, Fractional Schrödinger-Poisson systems with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud. 16 (2016), 15–30, https://doi.org/10.1515/ans-2015-5024. Search in Google Scholar
[56] J. Zhang, H. Zhou, and H. Mi, Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system, Adv. Nonlinear Anal. 13 (2024), 20230139, https://doi.org/10.1515/anona-2023-0139. Search in Google Scholar
[57] J. Zhang and Y. Zhang, An infinite sequence of localized semiclassical states for nonlinear Maxwell-Dirac system, J. Geometric Analysis 34 (2024), 277, https://doi.org/10.1007/s12220-024-01724-4. Search in Google Scholar
© 2025 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
- Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
- Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
- Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
- Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
- Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
- Existence results for non-coercive problems
- Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
- Geometric characterization of generalized Hajłasz-Sobolev embedding domains
- Subharmonic solutions of first-order Hamiltonian systems
- Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
- Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
- Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
- Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
- Homoclinic solutions in periodic partial difference equations
- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
- Properties of minimizers for L2-subcritical Kirchhoff energy functionals
- Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
- Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
- Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
- Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
- Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
- Existence of positive radial solutions of general quasilinear elliptic systems
- Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
- Sharp viscous shock waves for relaxation model with degeneracy
- Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
- Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
- Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
- Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
- Singularity for the macroscopic production model with Chaplygin gas
- Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
- Global dynamics of population-toxicant models with nonlocal dispersals
- α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
- High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- Boundedness of solutions to quasilinear elliptic systems
- Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
- Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
- Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
- Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
- Shape of extremal functions for weighted Sobolev-type inequalities
- One-dimensional boundary blow up problem with a nonlocal term
- Doubling measure and regularity to K-quasiminimizers of double-phase energy
- General solutions of weakly delayed discrete systems in 3D
- Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
- Optimal large time behavior of the 3D rate type viscoelastic fluids
- Local well-posedness for the two-component Benjamin-Ono equation
- Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
- Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
- Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
- On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
- Normal forms of piecewise-smooth monodromic systems
- Fractional Dirichlet problems with singular and non-locally convective reaction
- Sharp forced waves of degenerate diffusion equations in shifting environments
- Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
- Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
- Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
- Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
- Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
- Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
- Generalized quasi-linear fractional Wentzell problems
- Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
- Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
- Boundedness and global stability in a predator-prey chemotaxis system with indirect pursuit-evasion interaction and nonlocal kinetics
- Review Article
- Existence and stability of contact discontinuities to piston problems
Articles in the same Issue
- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
- Concentrating solutions for double critical fractional Schrödinger-Poisson system with p-Laplacian in ℝ3
- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
- Entire radial bounded solutions for Leray-Lions equations of (p, q)-type
- Combinatorial pth Calabi flows for total geodesic curvatures in spherical background geometry
- Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces
- Positive solutions for asymptotically linear Schrödinger equation on hyperbolic space
- Existence and non-degeneracy of the normalized spike solutions to the fractional Schrödinger equations
- Existence results for non-coercive problems
- Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent
- Geometric characterization of generalized Hajłasz-Sobolev embedding domains
- Subharmonic solutions of first-order Hamiltonian systems
- Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
- Stability of pyramidal traveling fronts in time-periodic reaction–diffusion equations with degenerate monostable and ignition nonlinearities
- Ground-state solutions for fractional Kirchhoff-Choquard equations with critical growth
- Existence results of variable exponent double-phase multivalued elliptic inequalities with logarithmic perturbation and convections
- Homoclinic solutions in periodic partial difference equations
- Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
- Properties of minimizers for L2-subcritical Kirchhoff energy functionals
- Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
- Blow-up solutions to the Hartree-Fock type Schrödinger equation with the critical rotational speed
- Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator
- Regularity for double-phase functionals with nearly linear growth and two modulating coefficients
- Uniform boundedness and compactness for the commutator of an extension of Riesz transform on stratified Lie groups
- Normalized solutions to nonlinear Schrödinger equations with mixed fractional Laplacians
- Existence of positive radial solutions of general quasilinear elliptic systems
- Low Mach number and non-resistive limit of magnetohydrodynamic equations with large temperature variations in general bounded domains
- Sharp viscous shock waves for relaxation model with degeneracy
- Bifurcation and multiplicity results for critical problems involving the p-Grushin operator
- Asymptotic behavior of solutions of a free boundary model with seasonal succession and impulsive harvesting
- Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds
- Stability of rarefaction wave for relaxed compressible Navier-Stokes equations with density-dependent viscosity
- Singularity for the macroscopic production model with Chaplygin gas
- Global strong solution of compressible flow with spherically symmetric data and density-dependent viscosities
- Global dynamics of population-toxicant models with nonlocal dispersals
- α-Mean curvature flow of non-compact complete convex hypersurfaces and the evolution of level sets
- High-energy solutions for coupled Schrödinger systems with critical growth and lack of compactness
- On the structure and lifespan of smooth solutions for the two-dimensional hyperbolic geometric flow equation
- Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
- On the existence of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations
- Remark on the analyticity of the fractional Fokker-Planck equation
- Continuous dependence on initial data for damped fourth-order wave equation with strain term
- Unilateral problems for quasilinear operators with fractional Riesz gradients
- Boundedness of solutions to quasilinear elliptic systems
- Existence of positive solutions for critical p-Laplacian equation with critical Neumann boundary condition
- Non-local diffusion and pulse intervention in a faecal-oral model with moving infected fronts
- Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals
- Qualitative properties of solutions to the viscoelastic beam equation with damping and logarithmic nonlinear source terms
- Shape of extremal functions for weighted Sobolev-type inequalities
- One-dimensional boundary blow up problem with a nonlocal term
- Doubling measure and regularity to K-quasiminimizers of double-phase energy
- General solutions of weakly delayed discrete systems in 3D
- Global well-posedness of a nonlinear Boussinesq-fluid-structure interaction system with large initial data
- Optimal large time behavior of the 3D rate type viscoelastic fluids
- Local well-posedness for the two-component Benjamin-Ono equation
- Self-similar blow-up solutions of the four-dimensional Schrödinger-Wave system
- Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation
- Existence and multiplicity of solutions for a class of superlinear p-Laplacian equations
- On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations
- Normal forms of piecewise-smooth monodromic systems
- Fractional Dirichlet problems with singular and non-locally convective reaction
- Sharp forced waves of degenerate diffusion equations in shifting environments
- Global boundedness and stability of a quasilinear two-species chemotaxis-competition model with nonlinear sensitivities
- Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian
- Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p-Laplacian
- Formation of singularities for a linearly degenerate hyperbolic system arising in magnetohydrodynamics
- Linear stability and bifurcation analysis for a free boundary problem arising in a double-layered tumor model
- Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system
- Generalized quasi-linear fractional Wentzell problems
- Existence, symmetry, and regularity of ground states of a nonlinear Choquard equation in the hyperbolic space
- Normalized solutions for NLS equations with general nonlinearity on compact metric graphs
- Boundedness and global stability in a predator-prey chemotaxis system with indirect pursuit-evasion interaction and nonlocal kinetics
- Review Article
- Existence and stability of contact discontinuities to piston problems
