In this paper, we are concerned with a class of Choquard equation with the lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. We emphasize that nonlinearities with doubly critical exponents are totally different from the well-known Berestycki–Lions-type ones. Working in a variational setting, we prove the existence, multiplicity and concentration of positive solutions for such equations when the potential satisfies some suitable conditions. We show that the number of positive solutions depends on the profile of the potential and that each solution concentrates around its corresponding global minimum point of the potential in the semi-classical limit.
Consider the following Choquard equation:
where , , is a parameter, is the Riesz potential, and . Note that and are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality, respectively. The potential function V satisfies the following conditions:
, ;
there exist points in such that is a strict global minimum, namely, , .
Condition expresses that V is a finite steep potential well. Assumptions of this type have been used in many recent papers for various types of elliptic problems; we refer only to Jeanjean–Tanaka [17] for the study of Schrödinger equation. Condition was used firstly used in [24] to study the existence of semiclassical states to Schrödinger–Poisson equation. Indeed, the similar condition as was used earlier in Cao–Noussair [6] where the authors studied the multiplicity of positive solutions. By using the change of variable , we can see that equation (
P
) is equivalent to the following one
When , equation (
P
˜
) is the following Choquard equation
For , and , the equation (
P
a
) goes back to the description of the quantum theory of a polaron at rest by Pekar in [32], and the modeling of an electron trapped in its own hole in the work of Choquard, as a certain approximation to Hartree–Fock theory of one-component plasma Penrose [33]. In recent years, equation (
P
a
) has been studied widely via variational and topological methods under various hypotheses on nonlinearity F. For the recent literature of the nonlinear Choquard equation with subcritical growth, we may turn to [18,22,25,26,30,34,41,44] and the references therein. We can also see [15,16,28,45] for the related problems involving only a single critical exponent growth, and see [19,20,36,37] for the case of double critical exponents growth.
Let us emphasize that Moroz–Van Schaftingen in [27] proved that there exists a ground state solution to equation (
P
a
), where the nonlinearity satisfies the general Berestycki–Lions-type assumptions:
there exists a constant such that for every ;
and ;
there exists a constant such that .
Cassani–Zhang [8] showed that there exists a ground state solution to equation (
P
a
), where the nonlinearity F satisfies and the following assumptions:
.
.
There exist and such that for there holds .
It is natural that there exist some choices of nonlinearity F such that conditions and do not work. One typical example for F is the so-called doubly critical exponents which is give by the following assumption:
and .
In particular, the authors [19,36,37,39] showed the existence of ground state solutions to equation (
P
a
) with , where F satisfies and
Recently, one general case for parameters N and α (that is, and ) was considered in [20] for Choquard equations with steep potential well and doubly critical exponents. The following case, to the best of our knowledge
has not been studied to problem (
P
a
) with finite potential well so far. The first purpose of the present work is to prove the existence of ground state solutions of equation (
P
a
) with . The first result reads as follows
Suppose that,,andhold. Letwhereandare defined in (
2.1
) and (
2.2
). Then the following statements hold true.
For any, equation (
P
a
) has at least a ground state solution.
for every, andis positive.
is radially symmetric decreasing with respect to 0.
In contrast to the case where , the case leads problem (
P
) to be more complicated in seeking weak solutions. The main obstacle arises in getting an exact estimate of the Mountain pass level which plays a key role in proving that Palais–Smale sequence is compact locally. The authors [19,36,37,39] used the Hardy–Littlewood–Sobolev inequality together with the Riesz potential estimate to establish an upper estimation of the Mountain pass level for case . Unfortunately, their fashions in [19,36,37,39] do not work in dealing with the case where . Although the authors in [20] investigated more general case where and , the steep potential well condition plays a significant role to overcome the difficulty arising form the loss of compactness due to the lower critical exponent. As a consequence, we have to establish some analysis which is different from those of the existed literature, to find an upper bound of Mountain pass level for case , and then use a generalization of Lions type theorem developed in [39] to prove the existence of positive ground state solutions.
The second purpose of this paper is to investigate the profile of positive solutions to (
P
) as . Indeed, in quantum physics one expects that as the Planck constant , the dynamic is governed by the external potential V and an interesting class of solutions show up which develop a spike shape around critical points of V. From the physical point of view, these solutions are known as semiclassical states, as they describe the transition from quantum mechanics to classical mechanics. For the detailed physical background, we refer to [31] and references therein. Wei–Winter [42] considered firstly the following Choquard equation
and, using a Lyapunov–Schmidt reduction method, they proved the existence of multibump solutions concentrating around local minima, local maxima or non-degenerate critical points of V. Secchi [35] considered (1.2) with V being a positive decaying potential and used a perturbation approach to prove the existence and concentration of bound states near local minima (or maxima) points of V as . Alves et al. [1] investigated the existence and concentration of solutions to equation (
P
) under V satisfying some local potential well conditions and F satisfying some conditions with superlinear and subcritical growth. Cassani–Zhang [8] showed the existence of ground state solution to equation (
P
) under conditions –. For related results see [2–4,11,14,23,46,47] and the references therein.
It is worthy of pointing out that Moroz–Van Schaftingen [29] established the existence of single-peak solution to equation (
P
) with , , and V satisfies
and there is an open bounded set O such that
which was firstly used in del Pino–Felmer [13] where the authors used the penalization approach to construct a single-peak solution to Schrödinger equation which concentrates around some locally minimum points of V. We emphasize that, in all the works mentioned above, the authors only studied the existence of semi-classical states to Choquard equation (
P
) with a power type nonlinearity or a general nonlinearity containing only the case which is a locally linear or superlinear linear problem at zero such that the exponential decay of solutions can be established by some classical technique (such as the standard Moser iteration technique and comparison principle and so on). However, the rest case can be seen as a locally sublinear case which results in that -regularity is not easy to get and then exponential decay of solutions can not be established. Thus, some classical penalization method used in the existed literature seems to be invalid in seeking concentrating solutions, because we can not prove that solutions of penalized problems are small in the penalized region. Based on the reasons above, Moroz–Van Schaftingen [29] set the rest case as an open problem. More precisely, a natural problem is
Recently, Cingolani–Tanaka [12] established the existence and multiplicity of single-peak solutions to equation (
P
), where potential V satisfies (), and F contains case . In particular, they developed a new tail minimizing operator together with the standard deformation argument (without building on the exponential decay of solutions) to prove the existence of concentrating solutions. However, it seems difficult to use their arguments to deal with the doubly critical case.
Very recently, Su et al. [39] used a nonlocal Brezis–Kato type regularity estimates in [27] and the decomposition of Riesz potential to establish -bound of solutions, and then to prove the existence and concentration of solutions to the fractional order version of equation (
P
) with and , where F satisfies with , and V satisfies the following condition:
() , and
where and are defined in inequalities (2.1) and (2.2). In , there is an extra restriction which plays a key role in their arguments.
Based on Theorem 1.1, we also prove the multiplicity of concentrating solutions to equation (
P
). The following is our second main result.
Suppose thatand–hold. Let N and α satisfy the following conditionThen the following conclusions hold:
There existssuch that for anyand, equation (
P
) possesses at least k positive ground state solutions,.
Eachhas a maximum pointwith
Compared with the results in [39], their methods can not be directly used to deal with our case, because the parameter α satisfies and the potential V just satisfies . Moreover, we also need to find a new Moser iteration technique to establish uniform bound of positive ground state solutions. Let us also point out that the arguments in [39] to multiple concentrating solutions are essentially based on the Ljusternik–Schnirelmann theory. In our multiple results, we know that the global maximum point of each positive solution concentrates around the associated global minimum point of the potential V, which makes concentration phenomenon become more specific than those in the literature.
This paper is organized as follows: In Section 2, we present some notations and inequalities. The existence and properties of positive solutions of the limit equation is proved in Section 3. In Sections 4–5, we give the proofs of multiplicity and concentration of bounded state solutions to equation (
P
), respectively. Moreover, C will be used repeatedly to denote various positive constants which may change from line to line.
Preliminaries
Define
with the semi-norm
Let
with the norm
Let us define
with the norm
is continuously embedded into for any and compactly embedded into for any . We recall the well-known Hardy–Littlewood–Sobolev inequality which will be used frequently later.
Letand. Ifis a bounded sequence insuch thata.e. inas, then
The limit problem
In this section, we consider the existence and properties of ground state solutions to the following equation
Equation (
P
a
) is variational and its solutions are the critical points of the functional defined in by
From Lemma 2.1, we can deduce that the functional . It is easy to see that if is a critical point of , i.e.,
for all .
Assume that the assumptions of Theorem
1.1
hold. Then the following conclusions hold:
For each, there exists a uniquesuch thatand, where
, whereand.
For all, we have, whereandMoreover, suppose thatand. Then u is a ground state solution for equation (
P
a
).
There exists a boundedsequencesuch that
We show the estimation of mountain pass level in the following lemma.
Assume that the assumptions of Theorem
1.1
hold. For any, we have
Let us define
then
Moreover,
It is easy to get that
Let . Then one has and . Hence, we know that for . Observe that
and
and
Let be the numbers satisfying
Then by defining , we see
which implies that
From (3.2), we obtain
from which we deduce that there exists independently of σ such that . Let , then . Indeed, we can also see easily from the above identity that . Hence, . We again pass to a upper limit as in (3.2) to obtain
which implies
We can choose small enough, such that . Set
It is easy to check that , for , and for . Hence, achieve its maximum at . That is to say, for any ,
Then, in virtue of the definition of functional and (3.1), we can always take small enough such that
This completes the proof. □
Assume that all assumptions of Theorem
1.1
hold. Letbe asequence ofwith. Then
Let be a bounded sequence. We show . Suppose on the contrary that
It then follows from Lemma 2.1 that
and
Based on the above facts, we get immediately
which imply that
Recalling (2.2) and (3.3), we have
which yields that
Combining (3.4) and (3.5), we have
This contradicts in Lemma 3.2. It remains to show . Suppose on the contrary that
Then, using again Hardy–Littlewood–Sobolev’s inequality, we obtain
and
Combining (3.6)–(3.8), we get
which imply that
It follows from (2.1) and (3.9) that
which yields that
Combining (3.10) and (3.11), we have
Recalling the definition of , we have for
This contradicts in Lemma 3.2. The proof is complete. □
In order to establish the existence of nonnegative ground state solutions to equation (
P
a
), we recall a Lions-type theorem established in [39] and a symmetrical theorem about ground states established in [27].
Assume thatand. Ifsatisfies, andis odd and has constant sign on, then every ground state of (
P
a
) has constant sign and is radially symmetrical with respect to some point in.
Existence. It is easy to see that is bounded in . According to Lemma 3.3, we get that and . In virtue of Proposition 3.1 there exists such that the sequence converges strongly in and a.e. in to . We now show that is a sequence. Indeed, it is easy to see that
For any , we obtain
where . It follows from that
Hence, is a nontrivial weak solution of equation (
P
a
). By Brezis–Lieb Lemma [5] and Lemma 2.4, we have
which implies and strongly in . Therefore, is a ground state solution of problem (
P
a
). Moreover, we could choose .
Positivity. Arguing similarly as in [38], we obtain for , and we can also obtain similarly for as in [39]. Hence, by using the elliptic regularity estimates and strong maximum principle, we can get easily.
Symmetric. From Lemma 3.4, we know that is radially symmetric with respect to some point . If , then is radially symmetric with respect to 0. □
Existence
In this section, we consider the existence of semi-classical state solutions to the following equation
whose energy functional defined in by
It is easy to verify that satisfies the mountain pass geometry structure. So, we define
where
Define
and . In order to study the effect of the shape of the graph of the potential V to the number of positive solutions of equation (
P
˜
), motivated by [9], we introduce the map
where is the Lebesgue measure of . Set
Define the function by
From [9], we know the map Φ is continuous in , if u is a radial function, and for .
For , let be the hypercube centered at , . and are the closure and the boundary of respectively. By assumptions and , we choose numbers such that are disjoint, for , and .
Let , and for , let
It is easy to show that and are non-empty sets for . Define for ,
Assume that all assumptions of Theorem
1.2
hold. There existsuch thatfor any,.
Let j be fixed. From Theorem 1.1, we see that there exists and such that
For small , we take such that
Set . In virtue of Lemma 3.1, there exists such that . By the properties of Φ, we have
By the Lebesgue dominate convergence theorem, we obtain
Then the continuity of Φ and implies . It follows from (4.2) that . Then we deduce that for small enough . Thus, for small ε.
Now we claim that . We first show that is bounded. In fact, if as , by , we obtain
Using the Lebesgue dominate convergence theorem, we also have
and
Then by (4.3), (4.4) and (4.5), we obtain a contradiction. Thus, up to a subsequence, we may assume that with . Furthermore, by , we obtain
If , then we have for ε sufficiently small,
which, together with (4.3), (4.5) and (4.6), implies by letting that . If , then we get directly. Letting in , we obtain . Since , so , which gives the desired assertion.
Consequently, by Lemma 3.1, (4.3), (4.5) and (4.6), we have
which concludes the proof. □
Assume that all assumptions of Theorem
1.2
hold. There existsuch thatfor all,.
For , arguing indirectly we assume that there exists a sequence such that . Then there exists a sequence such that . Note that , then by we obtain
Applying Lemma 3.1, there exists a unique such that , that is,
Then, by using Lemma 3.1 once again, we obtain
which implies that and
Moreover,
Applying the Ekeland’s variational principle [43], we deduce that there exists a sequence such that
According to Lemma 3.1, we have
is bounded in ;
;
.
Thus
Borrowing from Theorem 1.1, there exists a sequence such that and
and in . Hence,
By , we obtain . So, . We assume that . Thus, . By (4.8) and , we obtain
Moreover,
and in . By Fatou’s lemma,
It follows from (4.11) and (4.12) that . This contradicts , which completes the proof. □
Assume that all assumptions of Theorem
1.2
hold. For any, there existand a differentiable functiondefined for, andsuch that,andfor all, where.
The proof follows along the lines of [40, Lemma 2.4]. Define by
and
Since , we see that
and
Hence, applying the implicit function theorem at point , there exist and a differential function defined for such that , (4.13) holds and , which implies . Furthermore, by the continuity of functions Φ and t, we have . This completes our proof. □
Assume that the assumptions of Theorem
1.2
hold. For a fixed j, the valuehas a minimizing sequencesuch thatand, as.
Applying Ekeland’s variational principle [43] to (4.1), we have a minimizing sequence such that
for any . Applying Lemma 4.3 to , we obtain , function defined for , , such that . Choose . Let and , then by (4.14) we obtain
It follows from the mean value theorem that
Therefore,
On the other hand,
By (4.13) and the boundedness of , we see that there exists such that
Note that and as . Let in (4.15), we conclude
which implies . This completes the proof. □
Assume that all assumptions of Theorem
1.2
hold. For, letbe a sequence satisfyingand, for all, where δ,are defined in Lemma
4.1
. Thenhas a convergence subsequence in.
Note that the sequence is bounded in , we may assume that there exists such that weakly in , strongly in for every , and a.e. on . We will show that . Assume on the contrary that . By using and weakly in , for fixed, we obtain
Applying Lemma 3.1, there exists such that , that is
From , we obtain
This implies that at least one of the following results holds:
for some . Applying (4.17) and (4.19), we get that is bounded. We may assume that as , up to a subsequence. Using , (4.16), (4.17), and (4.18), one has
which implies by (4.19) that . So, by (4.16) we have
Let . Then, weakly in . Arguing as in the proof of Lemma 4.2 and taking into account (4.21), we obtain a sequence such that
It follows from (4.21) that
We know that is bounded in . Moreover, weakly in . If for all , up to a subsequence, such that
then according to the Loins vanishing lemma (see Lemma 2.2), we deduce that
Set . Then
Taking into account (4.23) and (4.25), we can infer that
It follows from , (2.1), (2.2), and (4.26) that
We define
both of which are finite since is bounded. Then we have
We now divide six cases to prove that (4.24) does not occur.
Case (1). If and , then we have due to (4.26). This contradicts .
Case (2). If and , then one has by (4.26), which implies . According to (4.23) and (4.25), we know that
On the other hand, by Lemma 4.1 and Lemma 3.2, there exist such that for any , . Then we obtain a contradiction.
Case (3). If and , then one has . According to (4.23) and (4.25), we have similarly that
Similarly to Case (2), we obtain a contradiction.
Case (5). If and , then according to (4.23) and (4.25), we have
Similarly to Case (3), we obtain a contradiction.
Case (6). If and , then according to (4.23) and (4.25), we have
Similarly to Case (3), we obtain a contradiction.
Summarizing Case (1)–(6), we know that there exist , and such that
Obviously, is unbounded. Set . Since is bounded in , we may assume that weakly in . It follows from (4.23) that
The weak convergence of implies that for any . Set . It follows from the Brezis–Lieb lemma [5] that
This together with (4.28) yields
We consider two cases: (i) as ; (ii) for some for large n.
Case (i). Without loss of generality, we assume that . It follows from the continuity of Φ that as . On the other hand, recalling (4.22), we have and , and then
for large enough n. It follows from that . This is a contradiction.
Case (ii) for some for large n. It follows from the Brezis–Lieb lemma [5] that
and
which implies that . Then there exists with as such that . And so, by (4.28) we have
It follows from and Lemma A that . By virtue of (4.30), we get , which contradicts with Lemma 4.1.
Hence, weakly in .
Now we show that strongly in . By , we obtain . Let . We claim that as . Otherwise, we assume that there exists such that . Arguing as in Case (ii), we obtain a contradiction. This completes the proof. □
For , Lemma 4.4 implies that has a minimizing sequence satisfying and as . It follows from Lemma 4.5 that satisfies (PS) condition, that is, strongly in , up to a subsequence. Then is a nontrivial solution of equation (
P
˜
). It is easy to see that solves equation (
P
˜
). By the strong maximum principle, we obtain in . Note that and strongly in . The we have
Lemmas 4.1 and 4.2 imply . Hence, we see that . Moreover, and are disjoint, . Therefore, equation (
P
˜
) has at least k distinct positive solutions. □
Concentration
In this section, for , are always referred as positive solutions of (
P
˜
) for fixed . We shall consider the concentration behavior of as .
Assume that the assumptions of Theorem
1.2
hold. Let. Then there exist a sequenceandsuch thatfor small.
Suppose by contradiction that there exists a sequence such that
for all . It follows from Lemma 2.2 that for any . We note that , then by Lemma 4.1 one has
Then arguing as in Lemma 4.5, we can infer that , which is a contradiction. This completes the proof. □
Assume that the assumptions of Theorem
1.2
hold. Forfixed,. Moreover,strongly in.
Let . We claim that is bounded in . Otherwise, . By Lemma 4.1, we have
which implies that is bounded in . Let , then by Lemma 5.1, we have weakly in . From , we have
Then, by recalling (4.16), we have
Then by , , Fatou’s lemma and Lemma A, we have
which implies by Lemma A that . Then, it follows from that strongly in . Hence
On the other hand, , that is,
This yields , which contradicts (5.1). Hence, is bounded in .
Without loss of generality, we assume that as . We are going to show that . Set . By Lemma 5.1, we have weakly in . Arguing similarly as above, we infer that strongly in , where solves
with . This together with the continuity of Φ and yields for small . Therefore, by Lemma A and , we obtain . This completes the proof. □
In the following, we will show the uniform bound of in for n and uniformly for . In order to get these results, we need the following lemmas.
Let. Then we havefor every. Moreover, there exists a constantindependent ofsuch that
Arguing similarly as [27, Proposition 3.1], we can obtain the conclusion. Here, we omit the details. □
Let. Then we have
Applying Lemma 5.3, we have that for every . Since
for every . Fix , can be decomposed as (see [10, Page 246, Lemma A.1]) , where and . Let in Lemma 2.3, then by , we have
Since , the following holds:
For , we have
Combining (5.3)–(5.5), we get and . Then . □
Let. For each, defineSetand(where) such thatFor, we have. Then we get
It follows from and that
for any . Let . From Lemma 5.4, one has
A direct calculation yields that
which gives
Notice that
Then, one has
and
and
Based on the above facts, we get
Combining (5.8) and (5.9), we have
Applying the Sobolev inequality, we get
Together (5.7), (5.10) with (5.11), we have
The proof is completed. □
Let. Then there existssuch that, anduniformly for.
It is easy to see that is bounded. Set .
Step 1. Using Lemmas 5.3–5.4 and [38, Lemma 3.9], and the Morse iterative technique, we obtain that there exists such that .
Step 2. We show that uniformly in . Putting and into (5.6), we get
Taking , we have
Let . From (1.3) in Theorem 1.2, we have that . Then . For any , we set
It is easy to see that
From Lemma 5.2, we have strongly in . Applying , we can choose and such that
Hence,
and
Then
Inserting (5.14) into (5.12), we deduce
which implies that
where , and .
Notice that . We can apply (5.15) with in place of , and and . It follows from and (5.16) that . Then, we have
and
Combining (5.13), (5.17) and (5.18), we can always choose such that
Then,
Similarly, we get
Iterating this procedure, for every integer m, we obtain
Set
For series , by using ratio test, we have
which implies that converges absolutely. Set
Then for series , by using root test, we get
which implies that converges absolutely. Passing to the limit as , putting (5.20)–(5.21) into (5.19), we obtain
Since strongly in , we have
Taking , we have
Combining (5.22)–(5.24), we know uniformly for . □
Assume that all assumptions of Theorem
1.2
hold. For, the functionhas a maximum pointsuch thatas.
We consider and a sequence as above. It is easy to see that is bounded in . Set . From Lemma 5.6, we know that there exists such that , and uniformly in . Repeat the arguments in Lemma 5.1, we know that there exist and such that
We claim that there exists such that
Otherwise, if , then
This gives a contradiction. As a consequence, (5.25) holds. This implies that there exists independent of such that
for small . Let be the maximum point of . By (5.26) and uniformly for , there exists such that . By the definition of , we obtain is the maximum point of . This together with Lemma 5.2 yields as . Hence, as . □
We have proven in the last section that equation (
P
˜
) admits at least k distinct positive solutions , . Then, are solutions of equation (
P
). By Lemma 5.7, is the maximum point of and satisfies as . The proof is complete. □
Footnotes
Acknowledgements
The first author would like to thanks Professor Van Schaftingen for his guidance about the decomposition of Riesz potential.
Y. Su is supported by the NSFC (No. 12101006); Z. Liu is supported by the NSFC (Nos. 11701267), the Hunan Natural Science Excellent Youth Fund (No. 2020JJ3029), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan, No. CUG2106211; CUGST2), and Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201900610).
Appendix
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