Double phase anisotropic variational problems involving critical growth
-
Ky Ho
and
Chao Zhang
Abstract
In this study, we investigate some existence results for double phase anisotropic variational problems involving critical growth. We first establish a Lions-type concentration-compactness principle and its variant at infinity for the solution space, which are our independent interests. Using these results, we obtain a nontrivial nonnegative solution to problems of generalized concave-convex type. We also obtain infinitely many solutions when the nonlinear term is symmetric. Our results are new even for the
1 Introduction
In this study, we investigate the existence of a nontrivial nonnegative solution and infinitely many solutions to the following double phase anisotropic problem of Schrödinger-Kirchhoff type:
where
Throughout this study, for
We make the following assumptions:
The function
There exist positive constants
Note that from
Moreover, as shown in [61, Eqn. (3)], there exist positive constants
The basic assumptions for the potential
The studies of differential equations and variational problems with nonhomogeneous operators and non-standard growth conditions have attracted extensive attention during the last few decades. The interest in the equations associated by nonhomogeneous nonlinearities has consistently developed in light of the pure or applied mathematical perspective to illustrate some concrete phenomena arising from nonlinear elasticity, plasticity theory, and plasma physics. Let us recall some related results by the way of motivation. Azzollini et al. [2,3] introduced a new class of nonhomogeneous operators with a variational structure:
where
which corresponds to the prescribed mean curvature operator defined by
In particular, Azzollini et al. in [2] proved the existence of a nontrivial nonnegative radially symmetric solution for the quasilinear elliptic problem:
where
with
where the nonlinearity
Anisotropic partial differential equations have recently gained significant attention, thanks to their applications in double- and multiphase variational energies, along with their relevance in integral form anisotropic energies. For insights into this topic, we refer the reader to the survey article [51] and references therein. Problem (1.5) is relevant to the double phase anisotropic phenomenon, in the sense that the differential operator has a different growth near zero and infinity. The double phase problems are described by the following functional:
with the so-called
This
In recent years, the study on partial differential equations with variable exponent has received an increasing deal of attention because they can be perceived as their application in the mathematical modeling of many physical phenomena occurring in diverse studies related to electro-rheological fluids, image processing, and the flow in porous media. There are many reference articles associated with the study of elliptic problems with variable exponent (see [25,27,58] and references therein for more background on applications). Also, we refer the readers to [24,41,57,59,61] for double phase differential problems with variable exponent. In this directions, Zhang and Rădulescu in a recent work [61] extended the results in [18] to the more general variable exponent case:
where the differential operator
where
This work can be seen as a continuation of the earlier works [16,59,61] to the case of Problem (1.1) with critical growth and containing a Kirchhoff term. The critical problem was originally studied in the pioneer study by Brezis–Nirenberg [13] dealing with Laplace equations. Since then, many researchers have been interested in such problems, and there have been extensions of [13] in many directions. As we know, one of the difficulties in studying elliptic equations in an unbounded domain involving critical growth is the absence of compactness arising in relation to the variational approach. To overcome this difficulty, the concentration-compactness principles (CCPs), which were initially provided by Lions [44,45], and its variant at infinity [8,10,17] have been used. In particular, these principles have played a crucial role in showing the precompactness of a minimizing sequence or a Palais-Smale sequence. By making use of these CCPs or extending them to the suitable solution spaces, many authors have been successful to deal with critical problems involving elliptic equations of various types (see e.g., [11,12,33,37–40] and references therein). Regarding the nonlocal Kirchhoff term, it was first introduced by Kirchhoff [42] to study an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Elliptic problems of Kirchhoff type have a strong background in several applications in physics and have been intensively studied by many authors in recent years (e.g., [1,19,26,31,60] and references therein).
The main feature of this article is to establish the existence and multiplicity of nontrivial nonnegative solutions to double phase anisotropic problem of Schrödinger-Kirchhoff type with critical growth. To do this, we first establish a Lions-type concentration-compactness principle and its variant at infinity for the solution space of (1.1). As mentioned earlier, as compared with the
As an application of this result, we will be concerned with the existence of a nontrivial nonnegative solution and infinitely many solutions to a class of double phase problems involving critical growth. Concerning this, on a new class of nonlinearities that is a concave-convex type of nonlinearities, we obtain the existence result of a nontrivial nonnegative solution via applying the Ekeland variational principle (Theorem 4.1). When the symmetry of the reaction term
This article is organized as follows. In Section 2, we review some properties of the variable exponent spaces. In Section 3, we establish the concentration-compactness principles for variable exponent Orlicz-Sobolev space, which plays as the solution space of Problem (1.1). Section 4 is devoted to the study of the existence and multiplicity of nontrivial solutions to Problem (1.1). In the Appendix, we provide a proof for a helpful inequality presented in Section 2.
2 Preliminaries and notations
2.1 Lebesgue-Sobolev spaces with variable exponent
In this subsection, we briefly recall the definition and some basic properties of the Lebesgue-Sobolev spaces with variable exponent, which were systematically studied in [27].
Let
and
For
endowed with the Luxemburg norm:
When
and
Some basic properties of
Proposition 2.1
[27] Define the modular
Then, for all
if
if
Furthermore, for
Proposition 2.2
[28,43] The space
Define the Sobolev space
endowed with the norm:
We recall the following result obtained by Fan (see [29, Theorem 1.1]).
Proposition 2.3
Let
Moreover, if
2.2 Variable exponent Orlicz-Sobolev spaces
In this subsection, we review the definition and properties of the variable exponent Orlicz-Sobolev space introduced in [61], which is the solution space for Problem (1.1) and the construction of energy functional.
Throughout this subsection, we assume
which is endowed with the norm:
Proposition 2.4
[61] The space
We define the variable exponent Orlicz-Sobolev space
which is equipped with the norm:
We have the following (see [61, Proposition 3.11 and Theorem 3.14 (ii)]).
Proposition 2.5
for any
Hereafter, we simply denote
Proposition 2.6
The following conclusions hold:
Assume additionally that
We also have the following compact imbedding, which is crucial to obtain our existence results; its proof can be found in [59, Theorem 3.10].
Proposition 2.7
Let
The next result is from [61, Corollary 3.13 (iii)] that shows the density of smooth functions with compact support in
Proposition 2.8
The space
Finally, we present a useful estimate regarding the main operator in (1.1), which is proved in Appendix and will be frequently used in the next sections. Define
and
Then, there exist positive constants
3 Concentration-compactness principles
Let
(see, e.g., [32, Section 1.3.3]). We identify
Let
We now state the main result of this section, i.e., a concentration-compactness principle for the variable exponent Orlicz-Sobolev space
Theorem 3.1
Let
Then, there exist
where
Note that the preceding result does not provide any information about a possible loss of mass at infinity. The next theorem expresses this fact in quantitative terms.
Theorem 3.2
Let
Then,
Assume, in addition, that:
(
Then,
Before giving a proof of Theorems 3.1 and 3.2, we review some auxiliary results for Radon measures.
Lemma 3.3
[39] Let
Lemma 3.4
[37] Let
for some constant
The following result is an extension of the Brezis-Lieb lemma to weighted variable exponent Lebesgue spaces.
Lemma 3.5
[39] Let
We are now in a position to prove Theorems 3.1 and 3.2. In the rest of this section, we denote the ball in
Proof of Theorem 3.1
Let
So, by Lemma 3.5, we deduce that
From this and (3.11), we easily obtain
i.e.,
By (2.3), it is clear that
for some finite nonnegative Radon measure
We will prove that
and
where
Assuming (3.15)–(3.17) for the moment, by taking the limit superior as
hence, (3.5) follows in view of Lemma 3.4 and (3.12).
To see (3.15), we first note that
From this and the facts that
Then, we obtain (3.15) by invoking Proposition 2.5 and (3.11).
To see (3.16), we note that
By this, we obtain (3.16) in view of Proposition 2.5.
Finally, we prove (3.17). To this end, we first note that by (1.4),
and
where
we obtain that
where
Up to a subsequence, we may assume that
Since
Thus, (3.17) immediately follows if we can show that
and
We only prove (3.22) since (3.23) can be proved similarly. Clearly, (3.22) holds for the case
Taking the limit as
Equivalently,
Thus, we have proved (3.22) (and similarly, (3.23)); hence, (3.17) follows due to (3.21).
We claim that
Thus, invoking Lemma 3.5 again, we infer
From this and the fact that
a contradiction. So,
Next, to obtain (3.7), let
Thus, by (3.1), we have
Taking the limit superior as
From Proposition 2.1, we have
It is clear that
Thus, we obtain
Hence,
On the other hand, arguing as that obtained (3.18), we obtain that
Invoking Proposition 2.1, we have
Similarly, we have
Thus, one has
where
for all
Now, we analyze the last term in (3.24) to obtain (3.7). First, we have
i.e.,
By Proposition 2.5 again, we have that
and
Also, we note that
Applying Proposition 2.2, using the fact
where
Hence, the estimate (3.34) infers
and
Hence, one has
From this and (3.32)–(3.33), we deduce from (3.31) that
Using (3.24), (3.25), (3.29), (3.30), and (3.35), we obtain (3.7).
Finally, to show (3.6), note that for any
Here and in the sequel, by
Thus,
Proof of Theorem 3.2
Let
For the first term of the right-hand side of (3.36), we note that
Thus, we obtain
For the second term of the right-hand side of (3.36), we note that
Clearly,
Using this and (3.37), we obtain (3.8) by taking the limit superior as
Arguing as earlier, we obtain
and using (3.40), we easily obtain (3.9) from (3.39). Moreover, by replacing
Next, we prove (3.10) when
Let
For
Using Proposition 2.1 with (3.43), we have
From this, (3.41), and (3.44), we deduce
On the other hand, arguing as that obtained (3.18) again with noting
Invoking Proposition 2.1 and taking into account (3.43) again, we have
Similarly, we have
and
From (3.46)–(3.49) and (3.37), we obtain
Now, we analyze the last term in (3.42) to show (3.10). First, we have
i.e.,
By Proposition 2.5, we have that
and
Also, we note that
Arguing as that leads to (3.35), we obtain
Hence, the estimate (3.54) infers
and
Equivalently,
From this, (3.52)–(3.53), we deduce from (3.51) that
Using (3.42), (3.45), (3.50), and (3.55), we arrive at
Letting
4 Application
As an application of Theorems 3.1 and 3.2, in this section we will obtain the existence of a nontrivial nonnegative solution and infinitely many solutions for Problem (1.1) when the reaction term is of generalized concave-convex type.
Throughout this section, let
with
Then, by Propositions 2.6 and 2.7, we have that
Assume, in addition, that
There exist
There exist
and
for a.e.
A typical example for
By a solution of Problem (1.1), we mean a function
This definition is clearly well defined under aforementioned assumptions thanks to the aforementioned imbeddings on
Our first existence result is the next theorem.
Theorem 4.1
(A nontrivial nonnegative solutions for the generalized concave-convex type problem) Let
In addition, if
Theorem 4.2
(Infinitely many solutions for the generalized concave-convex type problem) In addition to the assumptions of
Theorem 4.1, assume that
We have the following important remark.
Remark 4.3
In Theorems 4.1 and 4.2, if we additionally assume
We will make use of critical point theory to determine solutions to Problem (1.1). In order to obtain necessary properties regarding the Kirchhoff term, we truncate the function
Define
and
It is clear that
and
Let
and
where
for a.e.
and
From the Definition (4.4) of
In the next two subsections, we always assume that
Here and in the rest of this section,
4.1 Existence of a nontrivial nonnegative solution
In this subsection, we will prove Theorem 4.1 via using the Ekeland variational principle for
and we say that
where
Lemma 4.4
For any given
where
Proof
Let
for some
where
Since
where
It is not difficult to see that
and
for some positive constant
Set
and
We claim that
From this and (4.5), we have
We will show that
Noting the boundedness of
By (4.17) and
Finally, invoking the Young inequality,
for a positive constant
as
Arguing as that obtained (3.35), we obtain
Thus, (4.29) implies that
Since
By passing to the limit superior as
Combining this with (4.20) gives
Hence, one has
with
Next, we consider the other case, namely,
By the same way of deriving (4.26), (4.27), and (4.30), we obtain
Using this, (3.37), and (3.40) while passing to the limit superior as
Combining this with (3.10) gives
We obtain from the last two inequalities that
where
and
From the estimate (4.31) for the case
Arguing as those lead to (4.12), we have
Since
Passing to the limit as
where
Invoking Proposition 2.2, we have
where
where
Using this fact, we consider the following two cases.
Thus,
i.e.,
Thus,
i.e.,
Therefore, in any case, we obtain
where
This contradicts with (4.10), i.e., we have shown that
in view of Lemma 3.5. Moreover, we also have
in view of (4.1) and (4.17). Using (4.37), (4.38),
On the other hand, by the monotonicity of
This leads to
where
By the boundedness of
Using (4.39) and (4.41), we infer from (4.40) that
Invoking (4.17) again, we obtain from the last equality that
From this and (4.17) we derive
The next two lemmas provide several geometries of
where
Lemma 4.5
For each
where
Proof
Let
in view of (4.8) and (4.42). Thus, using
Noting
So, by taking
The proof is complete.□
Set
where
Lemma 4.6
Let
Proof
Let
Since
Proof of Theorem 4.1
Let
see the notations
where
Define
Then, from Lemma 4.6 and the definition of
Thus,
From (4.47) and (4.48), we produce a
Thus, up to a subsequence, we obtain that
Thus,
4.2 Existence of infinitely many solutions
In this subsection, we will prove Theorem 4.2 employing the genus theory for the truncated energy functionals. Our argument follows the proof of [19, Theorem 2.2].
Let
Thus,
where
Rewrite
where
and for any
Obviously, on
By (4.53), we find
where
for
Moreover, we have
and
Lemma 4.7
Let
for any
Proof
Let
which is a contradiction. Thus, we have
hence,
Next, we will construct the sequence of critical points
Lemma 4.8
Let
where
From the assumption of
Proof of Lemma 4.8
Let
For
i.e.,
Thus, by taking
and
we have
Now, for each
and
As in [30, Lemma 3.9], we have
Let
where
where
Lemma 4.9
For
where
Proof
Let
Thus,
Proof of Theorem 4.2
Let
From this and (2.3), we arrive at
and the proof is complete.□
-
Funding information: K. Ho was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Grant No. 2022R1A4A1032094). Y.-H. Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1F1A1057775). C. Zhang was supported by the National Natural Science Foundation of China (No. 12071098) and the Fundamental Research Funds for the Central Universities (No. 2022FRFK060022).
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript. KH and YK: writing – original draft preparation – reviewing and editing, CZ: writing – reviewing and editing.
-
Conflict of interest: Prof. Chao Zhang, who is the author of this article, is a current Editorial Board member of the journal Advances in Nonlinear Analysis. This fact did not affect the peer-review process.
-
Data availability statement: Data sharing is not applicable to this article as obviously no data sets were generated or analyzed during the current study.
Appendix A Proof of Inequality (2.3)
Proof of inequality (2.3)
We will prove that (2.3) holds true with
Let
where
For brevity, set
Thus, it holds
In this case, we obtain from (A1) that
Consequently, by invoking Proposition 2.1, we derive from (A1) that
If
If
Thus, by (A1) and Proposition 2.1, we obtain
If there are only two of
Thus, by (A1) and Proposition 2.1 again, we obtain
Finally, for
In any case, we obtain the following lower bound of
Next, we prove the upper bound of
From this and (1.4), we obtain
If
If
If
Thus, we know
If
Thus, one has
Thus, it holds
Applying Proposition 2.1 again, we drive from the last inequality that
Note that
If
If
In summary, we obtain the upper bound of
The proof is complete.□
References
[1] G. Autuori, P. Pucci, and M. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal. 196 (2010), no. 2, 489–516. 10.1007/s00205-009-0241-xSearch in Google Scholar
[2] A. Azzollini, P. d’Avenia, and A. Pomponio, Quasilinear elliptic equations in RN via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations 49 (2014), 197–213. 10.1007/s00526-012-0578-0Search in Google Scholar
[3] A. Azzollini, Minimum action solutions for a quasilinear equation, J. Lond. Math. Soc. 92 (2015), 583–595. 10.1112/jlms/jdv050Search in Google Scholar
[4] A. Bahrouni, V. D. Rădulescu, and D. D. Repovš, Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity 32 (2019), 2481–2495. 10.1088/1361-6544/ab0b03Search in Google Scholar
[5] P. Baroni, M. Colombo, and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121 (2015), 206–222. 10.1016/j.na.2014.11.001Search in Google Scholar
[6] P. Baroni, M. Colombo, and G. Mingione, Non-autonomous functionals, borderline cases and related function classes, St. Petersburg Math. J. 27 (2016), no. 3, 347–379. 10.1090/spmj/1392Search in Google Scholar
[7] P. Baroni, M. Colombo, and G. Mingione, Regularity for general functionals with double-phase, Calc. Var. Partial Differential Equations. 57 (2018), Paper no. 62, 48 pp. 10.1007/s00526-018-1332-zSearch in Google Scholar
[8] A. K. Ben-Naouma, C. Troestler, and M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains, Nonlinear Anal. 26 (1996), 823–833. 10.1016/0362-546X(94)00324-BSearch in Google Scholar
[9] F. Bernis and J. Garcia Azorero, I. Peral Alonso, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth-order, Adv. Differential Equations 1 (1996), 219–240. 10.57262/ade/1366896238Search in Google Scholar
[10] G. Bianchi, J. Chabrowski, and A. Szulkin, Symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent, Nonlinear Anal. 25 (1995), 41–59. 10.1016/0362-546X(94)E0070-WSearch in Google Scholar
[11] J. F. Bonder and A. Silva, Concentration-compactness principle for variable exponent spaces and applications, Electron. J. Differential Equations (2010), Paper No. 141, 18 pp. Search in Google Scholar
[12] J. F. Bonder, N. Saintier, and A. Silva, The concentration-compactness principle for fractional order Sobolev spaces in unbounded-domains and applications to the generalized fractional Brezis-Nirenberg problem, NoDEA Nonlinear Differential Equations Appl. 25 (2018), Paper No. 52, 25 pp. 10.1007/s00030-018-0543-5Search in Google Scholar
[13] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477. 10.1002/cpa.3160360405Search in Google Scholar
[14] H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8–9, 939–963. 10.1002/cpa.3160440808Search in Google Scholar
[15] S.-S. Byun and J. Oh, Regularity results for generalized double phase functionals, Anal. PDE 13 (2020), 1269–1300. 10.2140/apde.2020.13.1269Search in Google Scholar
[16] J. Cen, S. J. Kim, Y.-H. Kim, and S. Zeng, Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent, Adv. Differential Equations 28 (2023), no. 5–6, 467–504. 10.57262/ade028-0506-467Search in Google Scholar
[17] J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differential Equations 3 (1995), no. 4, 493–512. 10.1007/BF01187898Search in Google Scholar
[18] N. Chorfi and V. D. Rǎdulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 37 (2016), 12 pp. 10.14232/ejqtde.2016.1.37Search in Google Scholar
[19] N. T. Chung and K. Ho, On a p(⋅) -biharmonic problem of Kirchhoff type involving critical growth, Appl. Anal. 101 (2022), no. 16, 5700–5726. 10.1080/00036811.2021.1903445Search in Google Scholar
[20] F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. 195 (2016), 1917–1959. 10.1007/s10231-015-0542-7Search in Google Scholar
[21] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215 (2015), 443–496. 10.1007/s00205-014-0785-2Search in Google Scholar
[22] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), 219–273. 10.1007/s00205-015-0859-9Search in Google Scholar
[23] M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal. 270 (2016), 1416–1478. 10.1016/j.jfa.2015.06.022Search in Google Scholar
[24] Á. Crespo-Blanco, L. Gasiński, P. Harjulehto, and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differential Equations 323 (2022), 182–228. 10.1016/j.jde.2022.03.029Search in Google Scholar
[25] D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue spaces. Foundations and harmonic analysis, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Heidelberg, 2013. 10.1007/978-3-0348-0548-3Search in Google Scholar
[26] P. D’Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), no. 2, 247–262. 10.1007/BF02100605Search in Google Scholar
[27] L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer-Verlag, Heidelberg, 2011. 10.1007/978-3-642-18363-8Search in Google Scholar
[28] X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424–446. 10.1006/jmaa.2000.7617Search in Google Scholar
[29] X. Fan, Sobolev embeddings for unbounded domain with variable exponent having values across N, Math. Inequal. Appl. 13 (2010), 123–134. 10.7153/mia-13-09Search in Google Scholar
[30] G. M. Figueiredo and R. G. Nascimento, Multiplicity of solutions for equations involving a nonlocal term and the biharmonic operator, Electron. J. Differential Equations (2016), Paper No. 217, 15 pp. Search in Google Scholar
[31] A. Fiscella, G. Marino, A. Pinamonti, and S. Verzellesi, Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting, Rev. Mat. Complut. 37 (2024), 205–236. 10.1007/s13163-022-00453-ySearch in Google Scholar
[32] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer, New York, 2007. Search in Google Scholar
[33] Y. Fu, The principle of concentration compactness in Lp(x) spaces and its application, Nonlinear Anal. 71 (2009), 1876–1892. 10.1016/j.na.2009.01.023Search in Google Scholar
[34] Y. Fu and X. Zhang, Multiple solutions for a class of p(x)-Laplacian equations in involving the critical exponent, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466 (2010), no. 2118, 1667–1686. 10.1098/rspa.2009.0463Search in Google Scholar
[35] L. Gasiński and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differential Equations 268 (2020), 4183–4193. 10.1016/j.jde.2019.10.022Search in Google Scholar
[36] L. Gasiński and P. Winkert, Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold, J. Differential Equations 274 (2021), 1037–1066. 10.1016/j.jde.2020.11.014Search in Google Scholar
[37] K. Ho and Y.-H. Kim, The concentration-compactness principles for Ws,p(⋅,⋅)(RN) and application, Adv. Nonlinear Anal. 10 (2021), no. 1, 816–848. 10.1515/anona-2020-0160Search in Google Scholar
[38] K. Ho, Y.-H. Kim, and J. Lee, Schrodinger p(⋅)-Laplace equations in RN involving indefinite weights and critical growth, J. Math. Phys. 62 (2021), 1–28. 10.1063/5.0054557Search in Google Scholar
[39] K. Ho, Y.-H. Kim, and I. Sim, Existence results for Schrodinger p(⋅)-Laplace equations involving critical growth in RN, Nonlinear Anal. 182 (2019), 20–44. 10.1016/j.na.2018.12.004Search in Google Scholar
[40] K. Ho and I. Sim, On degenerate p(x)-Laplace equations involving critical growth with two parameters, Nonlinear Anal. 132 (2016), 95–114. 10.1016/j.na.2015.11.003Search in Google Scholar
[41] I. H. Kim, Y.-H. Kim, M. W. Oh, and S. Zeng, Existence and multiplicity of solutions to concave-convex-type double phase problems with variable exponent, Nonlinear Anal. Real World Appl. 67 (2022), Paper No. 103627, 25 pp. 10.1016/j.nonrwa.2022.103627Search in Google Scholar
[42] G. R. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanik, Teubner, Leipzig, 1876. Search in Google Scholar
[43] O. Kováčik and J. Rákosník, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41 (1991), 592–618. 10.21136/CMJ.1991.102493Search in Google Scholar
[44] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145. 10.1016/s0294-1449(16)30428-0Search in Google Scholar
[45] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. 10.4171/rmi/6Search in Google Scholar
[46] W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differential Equations 265 (2018), 4311–4334. 10.1016/j.jde.2018.06.006Search in Google Scholar
[47] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), 267–284. 10.1007/BF00251503Search in Google Scholar
[48] P. Marcellini, Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations 90 (1991), 1–30. 10.1016/0022-0396(91)90158-6Search in Google Scholar
[49] P. Marcellini, Regularity for elliptic equations with general growth conditions, J. Differential Equations 105 (1993), 296–333. 10.1006/jdeq.1993.1091Search in Google Scholar
[50] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 23 (1996), 1–25. Search in Google Scholar
[51] P. Marcellini, Anisotropic and p,q-nonlinear partial differential equations, Rend. Fis. Acc. Lincei 31 (2020), 295–301. 10.1007/s12210-020-00885-ySearch in Google Scholar
[52] G. Mingione and V. D. Rădulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl. 501 (2021), Paper No. 125197, 41 pp. 10.1016/j.jmaa.2021.125197Search in Google Scholar
[53] N. S. Papageorgiou, V. D. Rădulescu, and D. D. Repovš, Double phase problems and a discontinuity property of the spectrum, P. Amer. Math. Soc. 147 (2019), 2899–2910. 10.1090/proc/14466Search in Google Scholar
[54] N. S. Papageorgiou, V. D. Rădulescu, and D. D. Repovš, Existence and multiplicity of solutions for double phase Robin problems, B. London Math. Soc. 52 (2020), 546–560. 10.1112/blms.12347Search in Google Scholar
[55] K. Perera and M. Squassina, Existence results for double phase problems via Morse theory, Commun. Contemp. Math. 20 (2018), Paper no. 1750023, 14 pp. 10.1142/S0219199717500237Search in Google Scholar
[56] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291. 10.1007/BF00946631Search in Google Scholar
[57] M. A. Ragusa and A. Tachikawa, Regularity for minimizers for functionals of double phase with variable exponents, Adv. Nonlinear Anal. 9 (2020), 710–728. 10.1515/anona-2020-0022Search in Google Scholar
[58] V. D. Rădulescu and D. D. Repovš, Partial differential equations with variable exponents. Variational methods and qualitative analysis, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015. Search in Google Scholar
[59] X. Shi, V. D. Rădulescu, D. D. Repovš, and Q. Zhang, Multiple solutions of double phase variational problems with variable exponent, Adv. Calc. Var. 13 (2020), no. 4, 382–401. 10.1515/acv-2018-0003Search in Google Scholar
[60] P. Pucci, M. Xiang, and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785–2806. 10.1007/s00526-015-0883-5Search in Google Scholar
[61] Q. Zhang and V. D. Rădulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. 118 (2018), no. 9, 159–203. 10.1016/j.matpur.2018.06.015Search in Google Scholar
[62] S. D. Zeng, Y. R. Bai, L. Gasiński, and P. Winkert, Existence results for double phase implicit obstacle problems involving multivalued operators, Calc. Var. Partial Differential Equations 59 (2020), Paper No. 176, 18 pp. 10.1007/s00526-020-01841-2Search in Google Scholar
[63] S. D. Zeng, Y. R. Bai, L. Gasiński, and P. Winkert, Convergence analysis for double phase obstacle problems with multivalued convection term, Adv. Nonlinear Anal. 10 (2021), 659–672. 10.1515/anona-2020-0155Search in Google Scholar
[64] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710. Search in Google Scholar
[65] V. V. Zhikov, On Lavrentiev’s phenomenon, Russ. J. Math. Phys. 3 (1995), no. 2, 249–269. Search in Google Scholar
© 2024 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
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Articles in the same Issue
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
