In this paper, we study a class of the Kirchhoff type problems with convolution term in . With the appropriate assumptions on potential function V and convolution term f, together with the penalization techniques, Morse iterative method and variational method, the existence and multiplicity of multi-bump solutions are obtained for this problem. In some sense, our results also generalize some known results.
In this article, we deal with the following Kirchhoff type problems with convolution term in :
where , is the s-Laplacian operator, , , is the critical exponent, λ is a positive parameter. For potential function V and convolution term f satisfy the following assumptions:
such that ;
The bounded open set with smooth boundary , which possesses k disjoint bounded components:
and
Moreover, let f is a continuous function satisfies the following assumptions:
, for , and ;
there exists such that ;
there exists such that for all ;
the map is increasing for all .
Our study of problem (1.1) was motivated by two main reasons. On the one hand, when , problem (1.1) becomes the Kirchhoff problems, this problem have many important applications in physics. Indeed, Kirchhoff [18] introduced a model of the hyperbolic equation in physics
where ρ, , h, E, L are some physical parameter, which based on the classical wave equation. In recent years, many authors pay more attention to the existence of solutions for the Kirchhoff problem, please refer to [8,12,14,20,22,37] and reference therein. However, part of the inspiration for our study of problem (1.1) comes from Ding and Tanaka [10], who have studied problem (1.1) with . Specifically, they studied the following problem:
where if ; if . They proved problem (1.2) has at least bump solutions when . Subsequently, Alves and Yang [4] studied the following Schrödinger-Poisson system, and they proved the existence of positive multi-bump solutions by variational methods. Similar to the discussion in [10], the authors in [23] obtained the existence and multiplicity of bump solutions for a class of Kirchhoff type problems. Later, Alves and Figueiredo [2] discussed the general Kirchhoff problem
They also established the existence of bump solutions for the above problems by using variational methods. More interesting results, we refer to [3,16,17,21,21,24–26] and the references therein for more details.
On the other hand, when , problem (1.1) is driven by a differential operator with unbalanced growth. When problem (1.1) with becomes a -Laplacian equation
this problem has a rich physical background, since the double phase operator has been applied to describe steady-state solutions of reaction diffusion problems in biophysics, plasma physics, and chemical reaction analysis. Problem (1.3) originates from the following Choquard-Pekar equation:
Problem (1.4) was introduced by Pekar [36] to describe the quantum mechanics of a polaron. Early interesting results can be found in Lieb [27] and Lions [29]. Later, the authors in [30] proved some qualitative properties of positive solutions considering powers like . In [32], the authors proved some properties of positive solutions for a generalized Choquard equation. In [1], the authors obtained multiplicity and concentration of positive solutions for a class of Choquard equation. Once we turn our attention to the bump solution of the Choquard equations, we immediately see that the literature is relatively scarce, see [3,13,17]. In [3], Alves et al. studied the following Choquard equation in :
they proved that this equation has at least multi-bump solutions. Very recently, Zhang et al. [42] considered the singularly perturbed double phase problem with nonlocal Choquard reaction
where , , ϵ is a small positive parameter and V is the absorption potential. Combining variational and topological arguments from Nehari manifold analysis and Ljusternik-Schnirelmann category theory, they proved the existence of positive ground state solutions that concentrate around global minimum points of the potential V. Moreover, they also established the relationship between the number of positive solutions and the topology of the set where V attains its global minimum. For some other interesting results can be found in [5–7,11,33–35,39–41], and the references therein. However, we note that there is no results about the existence and multiplicity of multi-bump solutions for the Kirchhoff type problems with convolution term in .
Inspired by the above references, we prove in this paper the existence and multiplicity of bump solutions for the double phase Kirchhoff problems with convolution term in . As far as we know, these results have not been studied before. The main difficulty in dealing with problem (1.1) is the presence of several differential operators with unbalanced growth, which generates a double phase associated energy. To get over this difficulty, we need to build some new estimates for such a double phase Kirchhoff problems with convolution term, which are key points to apply this method. Moreover, the convolution term still brings some difficulties in proving the existence of bump solutions. Since, if we set with and . Obviously,
In order to overcome this difficulty, we use some ideas from Alves et al. [3,4]. In some sense, the results in this paper can be regarded as an extension of results in [3,4,10,23].
Our main results are the following.
Assume that,and–. For any non-empty subset Γ of, there existsuch that, problem (
1.1
) has a positive solution. Moreover, for thehas the following properties: for fix the subset Γ of, then for any sequence, we can extract a subsequencesuch thatconverges strongly into a function u, which satisfiesoutside, andis a least energy solution for the following problem:where.
Under the assumptions of Theorem
1.1
, there existssuch that for all, problem (
1.1
) has at leastnontrivial solutions.
The problem
In this section, we mainly dealt with studying the existence of least energy solution for problem . First, we give the following famous Hardy–Littlewood–Sobolev inequality, which will be used frequently.
Suppose thatandwith. Letand, then there exists a sharp positive constant, independent of f, h, such that
Let and
endowed with the norm
In this paper, we use the Banach space
endowed with the norm
where
Next, we intend to prove the existence of least energy solution for problem . To this end, let
Define
where and is the corresponding Nehari manifold defined by
Then, we mainly prove that there exists satisfies
In what follows, we set . Thus ,
and
with , .
Assume that–,–, letwith, then there exists an l-tuplesuch that, which meansand.
First, we choose with . Now, let , , then
It from the assumptions and that
Let , we then have . Thus
for some positive constants , and C. Since is a continuous function on and . This fact implies that there exists global maximum point .
Next, we will prove that is a strict concave function, which certify the above and .
Indeed, according to Riesz basis property of semigroups [28]:
Using the above property and , then we have
This fact implies that
To prove the strict concavity of , we should show that the following term is a convex function:
Obviously, and are convex functions in respect of . This fact implies that
has concave property, and it is obvious that the concavity of and . In fact, it follows from the assumption and we have
and
Therefore
is a strictly convex function. Thus, is a strictly concave function. Taking , then we have .
Next, we will prove was a nonzero number as . By contradiction, if and set , from –, we have
According to , we know that
Then
where small enough, this result contradicts the fact is the unique global maximum point of .
In order to prove this lemma, we should claim
Thus, for any , we can get
We note that we have the following conclusions:
In fact, if , we have . Thus
If . When , we know that . Since , we can get
Thus
If . When , we can get , Thus
From the above discussion, we have
or
Thus, by (2.3) and (2.4), we know that (2.2) holds.
Moreover, it follows from , and (2.4) that
where , constants. It follows from that
By means of the Sobolev inequality, we can get
Due to is nonzero, thus there exists satisfying
On the other hand, we note that
Therefore, we have . □
Assume that–,–, let bounded sequenceinwithin. Provided that, we have, hereandfor.
Without loss of generality, if . With the aid of assumptions –, together with the Hardy–Littlewood–Sobolev inequality and the Sobolev embeddings, we have
where ϵ and are some positive constants. We note that and in for . Then, one has
Furthermore, it follows from the that
which is an absurd. Taking a similar approach, we can also show that , for any . This completes the proof. □
Now, Let’s concentrate on proving the following technical theorem.
Assume that–,–, then problemhas a least energy solution u that is nonzero on each componentof.
First, from the Lemma 2.1, we can get a sequence of such that
Obviously, is bounded. Then, there exists such that in and in as , for . Based on the above discussion, we get that possesses a unique global maximum point and . Since , then , , and
Thus, we have
As discussion in Lemma 2.1, we have , . Thus, from Lemma 2.2, we have for any . Moreover, Lemma 2.1 implies that there exist satisfies
From in , Proposition 2.1 and Sobolev embedding theorem, we get
for . On the other hand,
Thus, by (2.6) and (2.8), we have
Next, we will prove that is a critical point for .
In fact, if we suppose that
Let small enough such that . Thus, we can choose some such that
In what follows, we fix and sufficiently small such that
where
By means of the deformation lemma (see [38]), we can find a continuous map
satisfies
and
where
Next, fixing , we have
In the sequel, setting
For any , by (2.9) and , there holds
Since
then
From the Claim 2.3 in [3] and (2.14), it is easy to know that there exists such that
and
which is absurd. Consequently, we show that is a critical point of . □
An auxiliary problem
In this section, we’re going to start by recalling that the energy functional associated with problem (1.1):
By condition , we know that the embedding is continuous for all . Therefore, is compactly embedded in for all .
Moreover, we consider an auxiliary problem using the thoughts found by del Pino & Felmer in [9]. By the conditions and means that there exists a unique such that . Now, we signing the numbers ξ and ν and introducing the functions
and
Thus
and
Now, fix a non-empty subset and for each , we fix a bounded open subset with smooth boundary satisfying
and the functions
where is the characteristic function on Γ. Obviously h satisfies
for any ;
uniformly in ;
for all ;
for each , ;
for any , ;
for each , the function is strictly increasing in and for each , the function is strictly increasing in .
Moreover, we consider the following auxiliary double phase Kirchhoff problem:
The energy functional corresponding to problem
(
P
)
λ
is given by
It is easy to show that . Moreover, we remark that a critical point of is solution of problem (1.1) if and only if in .
Theorem 1.1 and Corollary 1.2 can be restated as the following:
Assume that,and. Then, for any non-empty subset Γ of, there existssuch that for all, problem
(
P
)
λ
possesses a nontrivial solution. In addition, the familyhas the following properties: for any sequence, we can extract a subsequencesuch thatconverges strongly into a function u which satisfiesforand the restrictionis a least energy solution of
Under the assumptions of Theorem
3.1
, then, there existssuch that for all, problem
(
P
)
λ
has at leastnontrivial solutions.
condition for
In this section, we mainly show the compactness condition for . To do this, we first prove that the energy functional fulfills the mountain pass structure [38].
Let, the energy functionalhas the following properties:
there existsatisfyingif;
there existswithsatisfying.
In order to prove (i), from –, (2.2), the Hölder inequality and Hardy–Littlewood–Sobolev inequality, we get
Hence, there exist α, such that if .
As for (ii), we fix with .
Set
where
By , we have
Integrating this on with , we get
Thus
Since , let for t large enough, we can see that conclusion (ii) holds. □
Arguing as , we can find a positive constant d, independent of λ and ξ such that
Now, we can prove the following useful lemma:
Suppose thatis asequence ofwith, thenis bounded and there existssatisfyingfor each.
Let and . By , and (2.2), we have
This fact implies that sequence is bounded in . □
LetandThen there existssatisfying
It follows from and that
Then
Let () satisfy
This fact and the Hölder inequality imply that
due to
Similarly, we have
From (4.4)–(4.6), we know that (4.3) holds. Hence, the proof is completed. □
Suppose thatbe asequence towith. Then, for any, there exists, there holds
First, by Lemma 4.2, we know that there exists such that in and in for all as .
Next, fix such that . Take be a cut off function has following properties:
here . It follows from the sequence is bounded in that
Thus, it follows from the above inequalities (4.3) and imply that
So, we can make sufficiently large such that and the proof is complete. □
For any, the functionalsatisfies thecondition.
Let be a sequence for with . From Lemma 4.2, up to a subsequence, in in for all , for a.e. . Then, it is easy to obtain that .
Now, let fixed , according to the Sobolev’s compact embedding theorem, the subcritical growth of h and the Lebesgue Dominated Convergence Theorem, we have
Moreover, by Lemma 4.4, we know that
Hence, from the Sobolev embedding and the subcritical growth of h, we deduce that
So
we yield that
Together (4.7), (4.8) with (4.9), we have
In order to prove converges strongly to u in . For simplicity, for all , the linear functional L on defined by
for all and . Let . By the Hölder inequality, we have
Since in and as in . Obviously,
From (3.1)–(3.3) and Lemma 4.2, we also have
Since a.e. in and the Fatou’s lemma, we have
and
We note that
and
due to the function and are non-decreasing for . Thus, from (4.12)–(4.22), we have
It follows from (4.19)–(4.23) that
and
Then , this means that strong convergence in . Hence the proof is complete. □
The condition
In this section, we will study the behaviour of a sequence, that is, there is a sequence with , as , verifying
for some .
Suppose that sequencebe asequence for. Then, up to a subsequence, there existssuch thatin. Furthermore,
in;
in,for all, and u is a solution for
;
;
;
.
By Lemma 4.2, we can get is bounded in , and also sequence is bounded in . Therefore, up to a subsequence, there exists such that
Fix , let
Take , . Thus
From the Fatou’s lemma, we have
This fact implies that in . Thus, one has in .
As in , as the discussion in Proposition 5.1, we have
This fact implies that in .
Since and in , we have or, equivalently, , for . Moreover, the limit in combined with for implies that
proving that is a solution for the nonlocal problem .
Since
Conclusion (i) implies that
Let . From conclusion (i), we have
Then
From (iii),
This way
In the following, we adapt the arguments found in [19] to prove the boundedness outside for some solutions of problem .
Suppose thatbe a family of solutions forsuch thatin, as. Then, there existssuch that,. Hence,is a solution for problem (
1.1
) for.
First, we take be a sequence such that as and define . For any and , take the function as follows:
where . We note that γ is an increasing function, then it holds
Let
Fix such that . Then, by using the Jensen’s inequality, we have
Similarly, we can know the previous inequality is right for any . This means that
From (5.2),
Taking in (3.4). Then, we foucs on (5.3),
Since , we get
On the other hand, from the boundedness of and Lemma 4.2, thus there exists such that
From the growth assumptions on f, for any there exists such that
From (5.4), (5.5), (5.6) and (5.7), we have
where .
Let and fix . Obviously, , thus
As in , let R sufficiently large, we have that
Putting together (5.8), (5.9) and (5.10) we get
and let , we get .
Now, combining and the limit as in (5.8), we have
it follows from this fact that
We define as so that and . Then
Define
Using an iteration argument, there exists independent of m such that
Taking the limit as we obtain for all . Moreover, from Corollary 5.5 in [15], we know that for some (independent of n) and , with C independent of n. Since in , we have that uniformly in . □
In this section, we take with . Set , where is an open neighborhood of with if .
Now, we define the following energy functional:
which corresponds to the following Choquard equation
In what follows, we denote by the number given by
where
with and
In the same way, let
where
with and
As discussed in Section 2, it’s easy to see and such that
and
Next, we will show the relation between and .
The following conclusion holds:
for all;
.
(i) We note that . Obviously,
(ii) Acoording to the previous discussion, for each large enough, there exists with
Since is bounded in , there exists , subsequence of , such that converges and . As the same proof of Lemma 4.4, we can obtain that there exists such that
and
Moreover, we can get
and
From the definition of , we have
Then, together with conclusion (i), we know that
This completes the proof of Lemma 6.1. □
To prove our main conclusion, we take is the least energy solution of problem , i.e.,
Furthermore, from the Lemma 3.1, we know that there are small enough and large enough such that
and
where , .
Let
and denote by the class of continuous pathes which satisfies the following conditions:
and
where are the positive constants obtained in (6.2) and (6.3). As , it is easy to get . And by (6.5) for the path γ and (6.4), we get
From the above discussion, we can draw the following conclusions:
For any, there existssuch thatwhere,.
We note that on , we can obtain our main result by using of (6.2), (6.3) and the Miranda’s Theorem [31]. □
In the following, we are devote to get nonnegative solutions for large values of λ, which converges to a least energy solution of problem (1.1) as . Let
where r was given by (6.2) and τ is the positive constant satisfies
Moreover, we give the following set
Setting for small enough, and
Noting that
this fact implies that .
Let, there existandindependent of λ such that
In fact, if there exist and such that
From , we know that and are both bounded. Passing to a subsequence if necessary, we may assume that converges. By the Proposition 5.1, we know that there exists such that u is a solution for
with
Since , we have
Setting , we get
which yields , and . Thus, . However, by and , we can get , and so, . Thus,
for n large enough. So , which is a contradiction. Thus, we have completed the proof of the desired result. □
Now, we denoted the constants , as follows:
and
where δ was given in and
where R was fixed in . Furthermore, let
Assume thatandgiven in the previous proposition. Then, for, there exists a solutionof problem (
1.1
) such that.
Let . Assume that there are no critical points of in . It follows from verifies the condition with that there exists a constant satisfies
With the aid of Proposition 6.1, we get
where is small enough and it does not depend on λ.
Next, let is a continuous functional such that
and
Define the functional as follows:
where Y is a pseudo-gradient vector field for on . Observe that is well defined, once , for . The inequality
guarantees that the deformation flow defined by
verifies
and
We now consider two paths:
The path , .
As , we get
and
Once , , gives that
Furthermore, it is also easy to see that
Therefore
The path , .
Note that
and
for all . Moreover,
and
Therefore
is independent of λ and . Next, we assume that there exists such that
Now, we claim that
for large. In fact, letting , , if , from (6.10), we have
and we have nothing more to do. And so we assume that and set
Now, we will discuss two cases:
Case 1:, .
Case 2:, .
In the first case, we have and for all . Hence, from (6.10), we know
that is,
showing (6.13).
In the second case, we have the following situations:
There exists such that . Let , we have
since .
There exists such that . Let , we get
since .
for all , and there are such that for all with
From definition of , we have
then the mean value theorem implies that . Notice that
we can deduce that
which proves (6.13).
Therefore, let
we have that and
On the other hand, we can estimate
Since , it follows that
and
By (6.13) and (6.14), applying Lemma 6.2, we have
which leads to
this contradicts with the conclusion (ii) of Lemma 6.1. □
From the Proposition 6.2, we know that there exists with satisfying:
, ;
;
.
Thus, by Proposition 5.1, we have that converges in to a function , which satisfies outside Ω and , . In the sequel, we claim that in , for all . Infact, it is possible to prove that there is , which is independent of j, such that if u is a nontrivial solution of problem (1.1), then
However, the solution u verifies
showing that in , for all . This finishes the proof of Theorem 1.1. □
Footnotes
Acknowledgements
S. Shi was supported by NSFC (Grant No. 12271203) and Science and Technology Development Project of Jilin Province (Grant No. YDZJ202101ZYTS141).
Author declarations
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References
1.
C.A.Alves and M.Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasilinear Choquard equation via penalization method, Proc. Roy. Soc. Edinburgh Sect. A146 (2016), 23–58. doi:10.1017/S0308210515000311.
2.
C.O.Alves and G.M.Figueiredo, Multi-bump solutions for a Kirchhoff-type problem, Adv. Nonlinear Anal.5 (2016), 1–26. doi:10.1515/anona-2015-0101.
3.
C.O.Alves, A.B.Nóbrega and M.B.Yang, Multi-bump solutions for Choquard equation with deepening potential well, Cal Var Partial Differential Equations55(3) (2016), 48. doi:10.1007/s00526-016-0984-9.
4.
C.O.Alves and M.Yang, Existence of positive multi-bump solutions for a Schrödinger–Poisson system in , Discrete Contin. Dyn. Syst.36 (2013), 5881–5910. doi:10.3934/dcds.2016058.
5.
V.Ambrosio and V.D.Rădulescu, Fractional double-phase patterns: Concentration and multiplicity of solutions, J. Math. Pures Appl.142 (2020), 101–145. doi:10.1016/j.matpur.2020.08.011.
6.
V.Ambrosio and D.Repovš, Multiplicity and concentration results for a -Laplacian problem in , Z. Angew. Math. Phys.72 (2021), 1–33. doi:10.1007/s00033-020-01428-z.
7.
R.Bartolo, A.M.Candela and A.Salvatore, On a class of superlinear -Laplacian type equations on , J. Math. Anal. Appl.438(1) (2016), 29–41. doi:10.1016/j.jmaa.2016.01.049.
8.
S.Chen and L.Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on , Nonlinear Anal. Real World Appl.14 (2013), 1477–1486. doi:10.1016/j.nonrwa.2012.10.010.
9.
M.del Pino and P.L.Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations4 (1996), 121–137. doi:10.1007/BF01189950.
10.
Y.H.Ding and K.Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math.112 (2003), 109–135.
11.
G.M.Figueiredo, Existence of positive solutions for a class of elliptic problems with critical growth on , J. Math. Anal. Appl.378 (2011), 507–518. doi:10.1016/j.jmaa.2011.02.017.
12.
G.M.Figueiredo, Existence of positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl.401 (2013), 706–713. doi:10.1016/j.jmaa.2012.12.053.
13.
L.Guo and T.Hu, Multi-bump solutions for nonlinear Choquard equation with ootential wells and a general nonlinearity, Acta Math. Sci.40 (2020), 316–340. doi:10.1007/s10473-020-0202-x.
14.
X.He and W.Zou, Existence and concentration of positive solutions for a Kirchhoff equation in , J. Differential Equations252 (2012), 1813–1834. doi:10.1016/j.jde.2011.08.035.
15.
A.Iannizzotto, S.Mosconi and M.Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam.32 (2016), 1353–1392. doi:10.4171/RMI/921.
16.
C.Ji and V.D.Rădulescu, Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in , Manuscripta Math.164 (2021), 509–542. doi:10.1007/s00229-020-01195-1.
17.
C.Ji and V.D.Rădulescu, Multi-bump solutions for the nonlinear magnetic Choquard equation with deepening potential well, J. Differential Equations306 (2022), 251–279. doi:10.1016/j.jde.2021.10.030.
18.
G.Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
19.
G.Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. Math.14 (1989), 27–36. doi:10.5186/aasfm.1989.1425.
20.
Y.Li, F.Li and J.Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations253 (2012), 2285–2294. doi:10.1016/j.jde.2012.05.017.
21.
S.Liang, N.Chung and B.Zhang, Multi-bump solutions for fractional Schrödinger equation with electromagnetic fields and critical nonlinearity, Adv. Differential Equations25 (2020), 423–456. doi:10.57262/ade/1594692077.
22.
S.Liang, P.Pucci and B.Zhang, Multiple solutions for critical Choquard–Kirchhoff type equations, Adv. Nonlinear Anal.10 (2021), 400–419. doi:10.1515/anona-2020-0119.
23.
S.Liang and S.Shi, Existence of multi-bump solutions for a class of Kirchhoff type problems in , J. Math. Phys.54 (2013), 121510. doi:10.1063/1.4850835.
24.
S.Liang and S.Shi, On multi-bump solutions of nonlinear Schrödinger equation with electromagnetic fields and critical nonlinearity in , Calc. Var. Partial Differential Equations56(25) (2017), 1–29.
25.
S.Liang and J.Zhang, Existence of multi-bump solutions for a class of quasilinear Schrödinger equations in involving critical growth, Milan J. Math.83 (2015), 55–90. doi:10.1007/s00032-015-0236-z.
26.
S.Liang and J.Zhang, Multi-bump solutions for a class of Kirchhoff type problems with critical growth in , Topol. Methods Nonlinear Anal.48 (2016), 71–101.
27.
E.H.Lieb, Existence and uniqueness of the minimizing solution of Choquard nonlinear equation, Stud. Appl. Math.57 (1977), 93–105. doi:10.1002/sapm197757293.
28.
E.H.Lieb and M.Loss, Analysis, 2nd edn, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001.
29.
P.L.Lions, The Choquard equation and related questions, Nonlinear Anal.4 (1980), 1063–1072. doi:10.1016/0362-546X(80)90016-4.
30.
L.Ma and L.Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal.195 (2010), 455–467. doi:10.1007/s00205-008-0208-3.
31.
C.Miranda, Un’ osservazione Su un teorema di Brouwer, Bol. Un. Mat. Ital.3 (1940), 5–7.
32.
V.Moroz and J.Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal.265 (2013), 153–184. doi:10.1016/j.jfa.2013.04.007.
33.
N.S.Papageorgiou, A.Pudelko and V.D.Rădulescu, Non-autonomous -equations with unbalanced growth, Math. Ann. (2022). doi:10.1007/s00208-022-02381-0.
34.
N.S.Papageorgiou, D.Qin and V.D.Rădulescu, Nonlinear eigenvalue problems for the -Laplacian, Bull. Sci. Math.172 (2021), 103038. doi:10.1016/j.bulsci.2021.103039.
35.
N.S.Papageorgiou, V.D.Rădulescu and D.D.Repovs˘, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys.69(4) (2018), 108. doi:10.1007/s00033-018-1001-2.
36.
S.Pekar, Untersuchung Uber die Elektronentheorie der Kristalle, Akademie Verlag, 1954.
37.
J.Wang, L.Tian, J.Xu and F.Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations253 (2012), 2314–2351. doi:10.1016/j.jde.2012.05.023.
X.Xie, T.Wang and W.Zhang, Existence of solutions for the -Laplacian equation with nonlocal Choquard reaction, Appl. Math. Lett.135 (2023), 108418. doi:10.1016/j.aml.2022.108418.
40.
J.Zhang, W.Zhang and V.D.Rădulescu, Double phase problems with competing potentials: Concentration and multiplication of ground states, Math. Z.301 (2022), 4037–4078. doi:10.1007/s00209-022-03052-1.
41.
W.Zhang and J.Zhang, Multiplicity and concentration of positive solutions for fractional unbalanced double phase problems, J. Geom. Anal.32 (2022), 235. doi:10.1007/s12220-022-00983-3.
42.
W.Zhang, J.Zhang and V.D.Rădulescu, Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction, J. Differential Equations347 (2023), 56–103. doi:10.1016/j.jde.2022.11.033.
