1 Introduction and main results
This paper deals with the following class of nonlinear and nonlocal Schrödinger equations:
where is the dimensionalized Planck constant, , , F is the primitive function of f, is the Riesz potential defined for every by
and the external potential V satisfies:
When , , equation (1.1) reduces to the following nonlocal elliptic equation:
which is variational, in the sense that solutions of (1.2) turn out to be critical points of the energy functional
In particular, in the relevant physical case of dimension , and , (1.2) turns into the so-called Choquard equation
which goes back to the seminal work of Fröhlich  and Pekar , modeling the quantum Polaron and then used by Choquard  to study steady states of the one component plasma approximation in the Hartree–Fock theory . Equation (1.3) appears also in quantum gravity in the form of Schrödinger–Newton systems [51, 52, 53] in which a single particle is moving in its own gravitational field (self-gravitating matter), see also . Lieb in  proved the existence and uniqueness of positive solutions to (1.3) by using rearrangements techniques. Multiplicity results for (1.3) were then obtained by Lions [39, 40] by means of a variational approach. A class of solutions which turn out to be of great interest in Physics as well as Mathematics are minimal energy solutions, which were predicted by Pekar to have a stochastic characterization in terms of Brownian motion, a conjecture proved just thirty years later by Donsker and Varadhan [21, 22]. We refer to  and references therein for an extensive survey on the topic.
Set , and in (1.3):
Formally, as , equation (1.4) yields
which is a prototype in semilinear equations and in particular it is well known since the work of Gidas, Ni and Nirenberg  that positive solutions with finite energy are radially symmetric, unique and non-degenerate (in the sense that the kernel of the linearized operator at the solution u is generated by ), see [28, 49]. In contrast with the local problem (1.5), moving planes methods are somehow difficult to be used and is difficult to be used and the classification of positive solutions to (1.4) (even for ) has remained open for a long time. By using a suitable version of the moving planes method developed by Chen, Li and Ou , Ma and Zhao  gave a breakthrough to this open problem by considering equivalent Bessel–Riesz integral systems. By requiring some involved assumptions on α, p and N, they proved that positive solutions of (1.4) are, up to translations, radially symmetric and unique. In , Moroz and Van Schaftingen established the existence of ground state solutions to (1.4) in the optimal range
The endpoints in the above range of p are extremal values for the Hardy–Littlewood–Sobolev inequality  and sometimes called lower and upper H-L-S critical exponents. From the PDE point of view, a Pohozaev-type identity prevents the existence of finite energy solutions. In the upper critical case, as in the local Sobolev case, the appearance of a group invariance which yields explicit extremal functions to the H-L-S inequality is responsible for the lack of compactness. The lack of compactness can not be recovered by the presence of an external potential. In the lower critical case, equivalent variational characterizations of the ground state level still allow the H-L-S extremal functions to preventing compactness: this casts the problem within the class of Brezis–Nirenberg-type problems .
Recently, in  the more general Choquard equation (1.2) has been studied by requiring Berestycki–Lions-type conditions, and establishing the existence of ground state solutions in the subcritical case (1.6).
The first purpose of the present work is to investigate the existence of ground state solutions to (1.2) involving the upper H-L-S critical exponent. In presence of lower H-L-S critical exponent, a suitable external potential may lower down the groundstate to the compact region. This turns out to be a Lions-type problem and it is considered in a companion paper  as it involves quite different techniques.
Throughout this paper we assume which satisfies:
there exist and such that
Our first main result in this paper is the following:
Let us point out that assumption (F3) plays a crucial role. Indeed, under the lonely assumptions (F1) and (F2), equation (1.2) has no solutions for any nontrivial external potential V by means of a Pohozaev-type identity (Lemma 3.2, Section 2). This fact rules out any perturbative argument and casts the problem into a Brezis–Nirenberg-type.
The second purpose of this paper is to investigate the profile of positive solutions to (1.1) 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  and references therein. By a Lyapunov–Schmidt reduction approach, based on the non-degeneracy condition, in [23, 49] the authors obtained the existence of solutions to the semilinear singularly perturbed Schrödinger equation
which exhibit a single peak or multi peaks concentrating, as , around any given non-degenerate critical points of V. However, so far, the non-degeneracy condition holds for only a very restricted class of f. In the last decade, a lot of efforts have been devoted to relax or remove the non-degeneracy condition in this family of singularly perturbed problems. By using a variational approach, Rabinowitz  obtained the existence of positive solutions to (1.7) for small with the following global potential well condition:
Subsequently, by a penalization approach, del Pino and Felmer  weakened the above global potential well condition to the local condition
there exists a bounded domain such that
and proved the existence of a single-peak solution to (1.7). In [54, 18], the non-degeneracy condition is not required. Some related results can be found in [59, 19, 20, 17, 3] and the references therein. In  Byeon and Jeanjean introduced a new penalization approach and constructed a spike layered solution of (1.7) under (V2)* and the almost optimal Berestycki-Lions conditions , see also [11, 12, 9] and [65, 68].
The second main result of this paper is the following:
There exists a local maximum point of such that
and converges (up to a subsequence) uniformly to a ground state solution of the limit equation
for some .
We mention that related results under stronger assumptions have been recently obtained in . For the convenience of the reader let us better contextualize our result within the existing literature on the singularly perturbed problem (1.1).
In , Wei and Winter considered the nonlocal equation, equivalent to the Schrödinger–Newton system,
and by using a Lyapunov–Schmidt reduction method under assumption (V1), proved the existence of multi-bump solutions concentrating around local minima, local maxima or non-degenerate critical points of V. When the potential is allowed to vanish somewhere, thus avoiding (V1), the problem becomes much more difficult. In , Secchi considered (1.8) with a positive decaying potential and by means of a perturbative approach, proved the existence and concentration of bound states near local minima (or maxima) points of V as . Recently, by a nonlocal penalization technique, Moroz and Van Schaftingen  obtained a family of single spike solutions for the Choquard equation
around the local minimum of V as . In  the assumption on the decay of V and the range for are optimal. More recently, using the penalization argument introduced in , Yang, Zhang and Zhang  investigated the existence and concentration of solutions to (1.1) under the local potential well condition (V2)* and mild assumptions on f. In particular, the Ambrosetti–Rabinowitz condition and the monotonicity of are not required. For related results see [4, 58, 16, 43, 48, 56, 63, 7]. All the previous results are subcritical in the sense of the Hardy–Littlewood–Sobolev inequality. In , the authors considered the ground state solutions of the Choquard equation (1.1) in . By variational methods, the authors proved the existence and concentration of ground states to (1.1) involving critical exponential growth in the sense of the Pohozaev–Trudinger–Moser inequality. A natural open problem which has not been settled before is to establish concentration phenomena for (1.1) in the critical growth regime. Here we give a positive answer to this open problem in Theorem 1.2.
We conclude this section by giving the outline of the paper and pointing out major difficulties. In Section 2 we prove some preliminary results which require some efforts to extend a few well-known results in the local setting, to the nonlocal framework. Section 3 is devoted to proving Theorem 1.1. Here, without the Ambrosetti–Rabinowitz condition, to obtain the boundedness of the Palais–Smale sequence becomes a delicate issue. To overcome this difficulty, a possible strategy is to look for a constraint minimization problem. This goes back to Berestycki–Lions , in which the authors established the existence of ground state solutions to the scalar mean field equation . By using a similar strategy, Zhang and Zou  extended the result in  to the critical case. Precisely, in [6, 67], the existence of ground state solutions is reduced to looking at the constraint minimization problem
and eventually to get rid of the Lagrange multiplier thanks to some appropriate scaling. However, this approach fails for the nonlocal problem (1.2), since , and scale differently in space and hence one has no hope to remove the Lagrange multiplier. The existence of ground state solutions to the nonlocal problem (1.2), in the subcritical case, has been done by Moroz and Van Schaftingen in , where they constructed a bounded Palais–Smale sequence satisfying asymptotically the Pohozaev identity and obtained a ground state solution by virtue of a concentration-compactness-type argument and a scaling technique introduced by Jeanjean . Here, to avoid a Ambrosetti–Rabinowitz-type condition, we use the Struwe monotonicity trick, in the abstract form due to , to get a bounded Palais–Smale sequence. Clearly, due to the presence of a critical H-L-S term, the Palais–Smale condition fails. By a decomposition technique, we recover compactness and obtain the existence of ground state solutions to (1.2). In Section 4, we first prove some qualitative properties of the set of ground states such as compactness, regularity, symmetry and positivity. Then we use a truncation argument as key ingredient to prove Theorem 1.2. In , the authors considered problem (1.1) in the subcritical case and established concentration phenomena. Here, the presence of critical growth prevents to use directly the argument in . We overcome this difficulty by penalizing the problem which is relaxed to a subcritical case. The penalized problem admits a family of spike shaped solutions which develop a concentrating behavior around the local minima of V. Finally, the analysis carried out in Section 3 enables us to prove the convergence of the penalized solution to a solution of the original problem which preserves the same qualitative properties of the penalized problem.
In what follows, let be endowed with the norm
Before proving Theorem 1.1, we prove first some preliminary results. First of all, let us recall the following Hardy–Littlewood–Sobolev inequality which will be frequently used throughout the paper.
Lemma 2.1 ([37, Theorem 4.3]).
Let and with , and . Then there exists a positive constant (independent of ) such that
In particular, if , the best possible constant is given by
As a consequence of the Hardy–Littlewood–Sobolev inequality, for any , , . Moreover, and
2.1 Brezis–Lieb lemma
In this subsection, we prove a nonlocal version of the Brezis–Lieb lemma.
Lemma 2.2 (Brezis–Lieb Lemma).
Assume and there exists a constant such that
Let be such that weakly in and a.e. in as . Then
where as .
In order to prove Lemma 2.2, we recall the following lemma, which states that pointwise convergence of a bounded sequence implies weak convergence.
Lemma 2.3 ([62, Theorem 4.2.7]).
Let be a domain and let be bounded in for some . If a.e. in Ω as , then weakly in as
Proof of Lemma 2.2.
and there exists such that
which implies . For any sufficiently small, by the Hardy–Littlewood–Sobolev inequality there exists such that
Again by the Hardy–Littlewood–Sobolev inequality we have
where we have used the fact that is bounded in . It is easy to see there exists such that
Then, by Hölder’s inequality,
So for δ given above and fixed but large enough, we get for any n,
Similarly, let with large enough, we have
and for any n,
For , let . If , then we know that and for any . By noting that a.e. in Ω as , it follows from the Severini-Egoroff theorem that converges to u in measure in , which implies that as . Similarly we have, for n large enough,
Finally, let us estimate
where . Obviously, . By Lebesgue’s dominated convergence theorem we have
which implies by the Hardy–Littlewood–Sobolev inequality
as , and
as . Now let ; we have
and the arbitrary choice of δ concludes the proof. ∎
2.2 Splitting lemma
Next we prove a splitting property for the nonlocal energy.
Lemma 2.4 (Splitting Lemma).
where uniformly as for any .
Let be a domain and let be such that weakly in and a.e. in Ω as . Then the following hold:
For any and ,
Assume and as , for any , where . The following hold:
For any ,
If we further assume that , and , then
where uniformly for any as .
In the following, let C denote a positive constant (independent of ) which may change from line to line. For any fixed , there exists such that for . Choose such that
for . From the continuity of h, there exists such that for . Moreover, there exists such that
for . Noting that , we have . Then there exists such that
Setting ; then
Let . Then
Setting ; then and
Clearly, for ε given above, there exists such that
Noting that and as , there exists large enough, such that
Then, for any n,
Thus, by (2.1), for any n,
Finally, for given above, there exists such that
Recalling that weakly in , up to a subsequence, strongly in and there exists such that a.e. . Then we easily get for n large enough,
Moreover, let , then by (2.3),
By a.e. , we get as . Hence,
On the other hand, for ε given above, there exists such that
Noting that as , similarly as above, there exists such that for all n,
By the Lebesgue dominated convergence theorem,
where as uniformly in ϕ. Then by (2.5),
Then, by (2.4) and for n large,
Finally, combining the previous estimate with (2.2), we conclude the proof. ∎
Let , and let be bounded and such that, up to a subsequence, for any bounded domain , strongly in as . Then, up to a subsequence if necessary, a.e. in as .
Let us prove that for any fixed positive , passing to a subsequence if necessary, a.e. in as . Let be fixed and for any , there exists such that
Obviously, for any . Noting that , by Remark 2.1,
where the constant C depends only on . It follows that, up to a subsequence, strongly in and a.e. in as . Then, for almost every , one has
Since δ is arbitrary, the proof is completed. ∎
Now we are set to prove Lemma 2.4.
Proof of Lemma 2.4.
Observe that for any ,
Step 1. We claim
where uniformly for any as . Noting that , by Lemma 2.5 (ii) (1) with , ,
Then for , as well as and also , there exists such that
from which it follows
where uniformly for any as .
On the other hand, by virtue of (i) of Lemma 2.5 with and ,
For , as well as and also , one easily gets bounded in . By the Hardy–Littlewood–Sobolev inequality and Hölder’s inequality, there exists such that
where uniformly for any as . Then we get
which implies that
where uniformly for any as .
which yields, by Lemma 2.3, weakly in as . Noting that ,
and by Hölder’s inequality,
where uniformly for any as . The claim is thus proved.
Step 2. We claim
where uniformly for any as . The following hold:
where uniformly for any as . Let us only prove the first identity in (2.9), the remaining ones being similar. Observe that there exists and such that for and for . Noting that , we have . Then, for any , there exists (independent of ) such that
Then by the Hardy–Littlewood–Sobolev inequality and (2.6),
where uniformly for any as . So (2.9) holds.
Similarly we prove
where uniformly for any as . By the Hardy–Littlewood–Sobolev inequality and of Lemma 2.5, there exists such that
where uniformly for any as . So the first identity of (2.10) holds and the remaining can be proved in a similar fashion.
where uniformly for any as . To conclude the proof of (2.8), it remains to prove
where uniformly for any as . Notice that for any , there exist and such that for and for . Then, for any , by the Hardy–Littlewood–Sobolev inequality and Hölder’s inequality,
There exists (independent of ) such that
Then by (2.7), there exists (independent of ) such that
It follows that
3 Ground state solutions: Proof of Theorem 1.1
Since we are looking for positive ground state solutions to (1.2), we may assume that f is odd in . In this section, a key tool is a monotonicity trick, originally due to Struwe  and which here we borrow in the abstract form due to Jeanjean and Toland [34, 32].
For , we consider the following family of functionals:
Obviously, if f satisfies the assumptions of Theorem 1.1, for , and every critical point of is a weak solution of
The existence of critical points to is a consequence of the following abstract result
Theorem A (see ).
Let X be a Banach space equipped with a norm , let be an interval and let a family of -functionals be given on X of the form
Assume that for any , at least one between A and B is coercive on X and there exist two points such that for any ,
where . Then, for almost every , the -functional admits a bounded Palais–Smale sequence at level . Moreover, is left-continuous with respect to .
In the following, set and
and therefore for any and ,
On the other hand, for fixed and for any , by (F3),
and as . Then there exists (independent of λ) such that , and . Let
Here we remark that is independent of . In fact, let
where . Clearly, . On the other hand, for any , it follows from (3.2) that . Due to the path connectedness of , there exists such that if , if and . Then and
which implies that and so for any .
is bounded in ,
and in as .
Next, in the spirit of [33, 41], we establish a decomposition of such a Palais–Smale sequence , which will play a crucial role in proving the existence of ground states to (1.2). However, some extra difficulties with respect to the local case are carried over by the presence the nonlocal as well as critical H-L-S nonlinearity.
and in , ,
and as for any .
Before proving Proposition 3.1, we need a few preliminary lemmas.
Let and let be any nontrivial weak solution of (3.1). Then satisfies the following Pohozǎev identity:
Moreover, there exist (independent of ) such that and for any nontrivial solution , .
and this concludes the proof. ∎
Let . For any , combining the Hardy–Littlewood–Sobolev inequality with Sobolev’s inequality, we have
and it is achieved by the instanton
Now, we use this information to prove an upper estimate for .
Let , and assume
Then the following upper bound holds:
Let be a cut-off function with support such that on and on , where denotes the ball in of center at origin and radius r. Given , we set , where
where . Then we get
By direct computation, we know
and then by the Hardy–Littlewood–Sobolev inequality,
We also have
where for some , ,
Thus for some , we have
Here, we used the fact that . Then for any ,
One has as and for small. Following [55, Lemma 3.3], has a unique critical point in , which is the maximum point of . From ,
There exist (both independent of ε) such that for small.
where we used the fact that : hence a contradiction and . By (3.9), one has
By the Claim just proved and (3.8), we have for some ,
and hence on the one hand the following:
Then, for some ,
where we used the fact . Therefore, for any and small enough, we get
Proof of Proposition 3.1.
Let and assume weakly in and satisfy and in as .
Step 1. We claim in . As a consequence of Lemma 2.4, it is enough to show, up to a subsequence, that for any fixed ,
By virtue of the Hardy–Littlewood–Sobolev inequality and Rellich’s theorem, up to a subsequence, for some C (independent of n) we have
Step 2. Set ; we claim
where as . Next, we show that
Notice that and as , . It is easy to see that
which yields by the Hardy–Littlewood–Sobolev inequality that there exists some (independent of n) such that
Then by (3.11), we get
where as . Recalling that strongly in as , let
then . From
which is a contradiction. Thus (3.10) holds true.
Step 3. By (3.10) and weakly in as , there exists such that as and
Similarly as above, . Let . Then
If , i.e. strongly in as , then
and we are done. Otherwise, if strongly in as , similarly as above
Then there exists such that as and
Let . Then, up to a subsequence, weakly in as for some . We have and
Let . Then
If , i.e., strongly in as , then
and we are done. Otherwise, we can iterate the above procedure and by Lemma 3.2, we will end up in a finite number k of steps. Namely, let to have
Step 4. Clearly, as for . However, it is not clear that if repels each other as , i.e., as for any and . Let us show that after extracting a subsequence from and redefining if necessary, properties (iii), (iv), (v) hold. Let and satisfy and let be bounded if , whereas as if . Then, for any if , there exists such that, up to a subsequence, weakly in as and in . By Rellich’s theorem, for any , we have strongly in as . Noting that in and , similar to Step 2, we know that strongly in as . Then, up to a subsequence, there exists such that strongly in as , which eventually implies
Recalling that as , we have . Let and
Then similarly as above, up to a subsequence, for some , we have strongly in as . Then, as ,
Without loss of generality, we may assume that . Noting that a.e. in as , we get in . Then we redefine and as ,
By repeating the argument above at most times and redefining if necessary, we end up with such that
Finally, by Lemma 2.2 one has . The proof is now complete. ∎
Proof of Theorem 1.1.
As a consequence of Lemma 3.1, Proposition 3.1 and Lemma 3.2, one has that for almost every , problem (3.1) admits a nontrivial solution satisfying , , where (independent of λ). Then there exist and such that, as ,
By Pohozǎev’s identity (3.3) we have
and is bounded in . Notice that
Then by (3.13), up to a sequence, there exists such that
where we used the fact that is continuous from the left at λ. Moreover, by (3.13), for any ,
Similarly as above, there exists some such that
and by the Hardy–Littlewood–Sobolev inequality
where uniformly for any as . Namely, in as . Finally, we obtain
If strongly in , then , and in . Otherwise, as a consequence of Proposition 3.1 with , there exist and such that , in for all j and . So let
Then and .
We conclude the proof of Theorem 1.1 by showing that is achieved. Clearly, there exists such that as , and in . Thus is bounded in . Assume that weakly in as . Then in . If strongly in , then . Namely, is a ground state solution of (1.2). Otherwise, there exist and such that , in for all j and . By the definition of , , and , which yields as a ground state solution to (1.2). The proof is now complete. ∎
4 Towards semiclassical states
4.1 Compactness of the set of ground state solutions
Denote the set of ground state solutions to (1.2) by
Then by Theorem 1.1, for any . Since is invariant by translations, cannot be compact in . However, this turns out to be the only way to loose compactness as we have the following result.
For any , up to translations, is compact in .
Let . Then and in . Similarly as above is bounded in . Assume that weakly in as ; then in . If strongly in , we are done. Otherwise, by virtue of Proposition 3.1, up to a subsequence, there exists , and such that , in for all j and
which implies that , , and as . ∎
4.2 Regularity, positivity and symmetry
Let . The following hold:
For any , for .
For any , u has constant sign and is radially symmetric about a point.
coincides with the mountain pass value.
There exist , independent of , such that , , for , where .
First, by Pohozaev’s inequality it follows that is bounded in .
For any , there exists such that
In fact, for any fixed , let and in . Let and be such that for , for and for . Set
Note that are uniformly bounded in and so are in for any . Thanks to the compactness of , for any we can choose R depending only on ε such that
The map is uniformly bounded in for all .
Thanks to (4.1), for some c (independent of u) such that for any ,
As in [64, Proposition 2.2], we can choose with and with , and there exist (independent of u) such that
which combining with (4.1) implies the claim.
where . Following , for any , there exists a path such that and achieves its maximum at . Thereby, . Namely, is also a mountain pass value. Moreover, for any , u has a constant sign and is radially symmetric about some point. If u is positive, then u is decreasing at , where is the maximum point of u. Finally, by the radial lemma, uniformly as for . By the comparison principle, there exist , independent of such that for . ∎
4.3 Proof of Theorem 1.2
Let , and consider the following problem:
Let be the completion of with respect to the norm
For any set and , we define and . Since we are looking for positive solutions of (1.1), from now on, we may assume that for . For , let
Fix an arbitrary and define
as well as
Let be given by
To find solutions of (4.2) which concentrate inside O as , we look for critical points of satisfying . The functional that was first introduced in  will act as a penalization to forcing the concentration phenomena inside O. In what follows, we seek the critical points of in some neighborhood of ground state solutions to (1.2) with .
4.4 The truncated problem
The set is compact in .
By Proposition 4.2, . For any , without loss of generality, we assume that weakly in and a.e. in as . Let us first prove that . Indeed, by (v) of Proposition 4.2, there exist (independent of n) such that for any . By the Lebesgue dominated convergence theorem, strongly in as for any . So if , one has strongly in as , which contradicts the fact . We claim strongly in as . Indeed, if not, by Proposition 3.1, there exist and such that , in for all j and . Noting that and , we get a contradiction. Finally, . Clearly, is positive and radially symmetric. Recalling that 0 is the same maximum point for any n, by the local elliptic estimate, 0 is also a maximum point of . The proof is complete. ∎
By Proposition 4.2, let be fixed and satisfy
For fixed, let and consider the truncated problem
whose associated limit problem is
We have .
4.5 Proof of Theorem 1.2
In the following, we use the truncation approach to prove Theorem 1.2. First, we consider the truncated problem (4.4). By Lemma 4.2, is a compact subset of . Inspired from  we show that (4.4) admits a nontrivial positive solution in some neighborhood of for small ε. Then we show that there exists such that
As a consequence, turns out to be a solution to the original problem (1.1).
For this purpose, set
Let and consider a cut-off such that , for and for . Set , , and for some and , we define
In the following, we show that (4.4) admits a solution in for small enough, where
where . As in Proposition 4.2, is uniformly bounded in for all ε. Then, noting that for all , local elliptic estimates (see ) yield as . It follows from (4.3) that uniformly for small . Therefore, for small , , , and then is a positive solution to (1.1). ∎
Conflict of interest.
The authors declare they have no conflict of interest.
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About the article
Published Online: 2018-06-07
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1184–1212, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0019.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0