Existence of groundstates for a class of nonlinear Choquard equations

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- \Delta u + u = \bigl(I_\alpha \ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_\alpha) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\infty)), then (u) is of constant sign and radially symmetric.


Introduction
We consider the problem where N ≥ 3, α ∈ (0, N ), I α : R N → R is the Riesz potential defined for every x ∈ R N \ {0} by I α (x) = Γ( N −α 2 ) Γ( α 2 )π N/2 2 α |x| N −α , F ∈ C 1 (R; R) and f := F ′ . Solutions of (P) are formally critical points of the functional defined by We are interested in the existence of solutions to (P). Problem (P) is a semilinear elliptic problem with a nonlocal nonlinearity and covers in particular for N = 3, α = 2 and F (s) = s 2 2 the Choquard-Pekar equation [16,25], also known as the stationary Hartree equation or the Newton-Schrödinger equation [23]. In this case the existence of solutions was proved by variational methods by E. H. Lieb, P.-L. Lions and G. Menzala [16,18,21] and also by ordinary differential equations techniques [7,23,27]. In the more general case F (s) = s p p , problem (P) is known to have a solution if and only if N +α N −2 < p < N +α N [20, p. 457; 24, theorem 1] (see also [10,Lemma 2.7]).
The existence results up until now were only available when the nonlinearity F is homogeneous. This situation contrasts with the striking existence result for the corresponding local problem which can be considered as a limiting problem of (P) when α → 0, with g = F f . H. Berestycki  where G(s) = s 0 g(σ) dσ (and if g = F f , then G = F 2 2 ). They also proved that if u ∈ L ∞ loc (R N ) is a finite energy solution of (1.1) then u satisfies the Pohožaev identity [3, Proposition 1] This, in particular, implies that assumptions (g 1 ), (g 2 ) and (g 3 ) are "almost necessary" for the existence of nontrivial finite energy solutions of (1.1). Indeed, the necessity of (g 3 ) follows directly from (1.2). For (g 1 ) and (g 2 ), if f (s) = s p with s ∈ (1, N +2 N −2 ) then (1.2) immediately implies that (1.1) does not have any bounded finite-energy nontrivial solution.
In this spirit, we prove the existence of solutions to Choquard equation (1.1), assuming that nonlinearity f ∈ C(R; R) satisfies the growth assumption: there exists C > 0 such that for every s ∈ R, |sf (s)| ≤ C |s| We call any weak solution u of (P) which lies in the Sobolev space is a weak solution of (P) .
Similarly to assumptions (g 1 ), (g 2 ) and (g 3 ) in the case of the local problem (P), our assumptions (f 1 ), (f 2 ) and (f 3 ) are "almost necessary" for the existence of nontrivial solutions to (P). Indeed, by the Pohožaev identity (proposition 3.5), any solution u ∈ H 1 (R N ), [24, theorem 2]). If (f 3 ) is not satisfied, then a solution u ∈ H 1 (R N ) would satisfy −∆u + u = 0 and then necessarily be trivial.
In the limit α → 0, the assumptions (f 1 ), (f 2 ) and (f 3 ) do not allow to recover exactly (g 1 ), (g 2 ) and (g 3 ). The gap between (f 1 ) when α → 0 and (g 1 ) is purely technical. When α → 0, (f 2 ) gives the assumptions lim s→0 F (s) 2 /|s| 2 = 0 and lim |s|→∞ F (s) 2 /|s| 2N/N −2 = 0, which is stronger than (g 2 ). The first assumption is not really surprising, as it can be observed that in (1.1) both g (u) and u have the same spatial homogeneity and therefore by scaling it could always be assumed that lim s→0 G(s)/s 2 = 0. The second assumption is equivalent to lim sup |s|→∞ F (s) 2 /|s| 2N/N −2 ≤ 0. Finally (f 3 ) gives G(s) = F (s) 2 ≥ 0, which is actually weaker than (g 3 ). This weakening of the condition can also be explained by the difference between the various scalings of the problem (P).
We also obtain some qualitative properties of groundstates of (P).
Theorem 2 (Qualitative properties of groundstates). Assume that N ≥ 3, α ∈ (0, N ) and f ∈ C(R; R) satisfies (f 1 ). If f is odd and has constant sign on (0, ∞) then every groundstate of (P) has constant sign and is radially symmetric with respect to some point in R N .
Before explaining the proof of theorem 1, we recall the strategy of the proof of H. Berestycki and P.-L. Lions of the existence of solutions to (1.1) [3, §3]. They consider the constrained minimization problem (1.5) min they first show that by the Pólya-Szegő inequality for the Schwarz symmetrization, the minimum can be taken on radial and radially nonincreasing functions. Then they show the existence of a minimum v ∈ H 1 (R N ) by the direct method in the calculus of variations. This minimum v satisfies the equation with a Lagrange multiplier θ > 0. They conclude by noting that The approach of H. Berestycki and P.-L. Lions fails for (P) for two different reasons. First, the nonlocal term will not be preserved or controlled under Schwarz symmetrization unless the nonlinearity f satisfies the more restrictive assumption of theorem 2. Second, the final scaling argument fails: the three terms in (P) scale differently in space, so one cannot hope to get rid of a Lagrange multiplier by scaling in space.
In order to prove the existence of solutions in section 2, instead of the constrained minimization problem of type (1.5), we consider the mountain pass level where the set of paths is defined as Classically, in order to show that b is a critical level of the functional I, one constructs a Palais-Smale sequence at the level b, that is, a sequence (u n ) n∈N in H 1 (R N ) such that I(u n ) → b and I ′ (u n ) → 0 as n → ∞. Then one proves that the sequence (u n ) n∈N converges up to translations and extraction of a subsequence [26,32]. The first step of this approach is to establish the boundedness of the sequence (u n ) n∈N in H 1 (R N ). Usually this involves an Ambrosetti-Rabinowitz type superlinearity assumption, which in our setting would require the existence of µ > 1 such that s ∈ R + → F (s)/s µ is nondecreasing. In order to avoid introducing an Ambrosetti-Rabinowitz type condition, in section 2 we employ a technique introduced by L. Jeanjean, which consists in constructing a Palais-Smale sequence that satisfies asymptotically the Pohožaev identity [12] (see also [11]). This improvement is related to the monotonicity trick of M. Struwe [26,§II.9] and L. Jeanjean [13]. This allows to prove the existence of a nontrivial solution u to (P) under the assumptions (f 1 ), (f 2 ) and (f 3 ) only. A novelty in our proof, apart of the presence of the nonlocal term in the equation, is that we combine L. Jeanjean method with a concentration-compactness argument.
To conclude that such constructed solution u is a groundstate, we first show I(u) = b. This is a straightforward computation if u satisfies the Pohožaev identity (1.4) proved in section 3.3. This however brings a regularity issue, as the proof of the identity (1.4) requires a little more regularity than u ∈ H 1 (R N ). The subcriticality assumption (f 1 ) is too weak for a direct bootstrap argument. Thus we study the regularity of u in section 3.1 by a variant of the Brezis-Kato regularity result [5]. The relationship between critical levels b and c is established with the construction of paths associated to critical points in section 4.1 following L. Jeanjean and H. Tanaka [14].
The qualitative properties of the groundstate of theorem 2 are established in section 5. We show that the absolute value of a groundstate and its polarization are also groundstates. This leads to contradiction with the strong maximum principle if the solution is not invariant under these transformations.
Finally in section 6 we explain how the proof of theorem 1 can be simplified under the assumptions of theorem 2 using symmetric mountain pass [29], adapting the original argument of Berestycki and Lions for (P).

Construction of a solution
2.1. Construction of a Pohožaev-Palais-Smale sequence. We first prove that there is a sequence of almost critical points at the level b defined in (1.6) that satisfies asymptotically (1.4). We define the Pohožaev functional P : Proof. Our strategy consists in first proving in claims 1 and 2 that the functional I has the mountain pass geometry before concluding by a minimax principle.

Claim 1. The critical level satisfies
Proof of the claim. We need to show that the set of paths Γ is nonempty. In view of the definition of Γ, it is sufficient to construct u ∈ H 1 (R N ) such that I(u) < 0. If we choose s 0 of assumption (f 3 ) so that F (s 0 ) = 0 and set w = s 0 χ B1 , we obtain We will take the function u in the family of functions On this family, we compute for every τ > 0, and observe that for τ > 0 large enough, Proof of the claim. Recall the Hardy-Littlewood-Sobolev inequality [17, theorem 4 where C > 0 depends only on α, N and s. By the upper bound (f 1 ) on F , for every u ∈ H 1 (R N ), Hence there exists δ > 0 such that if R N |∇u| 2 + |u| 2 ≤ δ, then and therefore Since γ ∈ Γ is arbitrary, this implies that b ≥ δ 4 > 0. ⋄ Conclusion. Following L. Jeanjean [12, §2] (see also [11, §4]), we define the map Φ : For every σ ∈ R and v ∈ H 1 (R N ), the functional I • Φ is computed as In view of (f 1 ), As Γ = {Φ •γ :γ ∈Γ}, the mountain pass levels of I and I • Φ coincide: By the minimax principle [32, theorem 2.9], there exists a sequence ( we reach the conclusion by taking u n = Φ(σ n , v n ).

Convergence of Pohožaev-Palais-Smale sequences.
We will now show how a solution of problem (P) can be constructed from the sequence given by proposition 2.1.
such that I ′ (u) = 0 and a sequence (a n ) n∈N of points in R N such that up to a subsequence u n (· − a n ) ⇀ u weakly in H 1 (R N ) as n → ∞.
Proof. Assume that the first part of the alternative does not hold, that is, We first establish in claim 1 the boundedness of the sequence and then the nonvanishing of the sequence in claim 2.
Proof of claim 1. For every n ∈ N, As the right-hand side is bounded by our assumptions, the sequence (u n ) n∈N is bounded in Proof of the claim. First, by (2.2) and the definition of the Pohožaev functional P we have For every n ∈ N, the function u n satisfies the inequality [19, lemma I.1; 24, lemma 2.3; 32, lemma As F is continuous and satisfies (f 2 ), for every ǫ > 0, there exists C ǫ such that for every s ∈ R, and the Hardy-Littlewood-Sobolev inequality implies that

Conclusion.
Up to a translation, we can now assume that for some p ∈ (2, 2N By Rellich's theorem, this implies that up to a subsequence, (u n ) n∈N converges weakly in Since the sequence (u n ) n∈N converges weakly to u in H 1 (R N ), it converges up to a subsequence to u almost everywhere in R N . By continuity of F , (F • u n ) n∈N converges almost everywhere to F • u in R N . This implies that the sequence (F • u n ) n∈N converges weakly to F • u in L 2N N +α (R N ). As the Riesz potential defines a linear continuous map from L On the other hand, in view of (f 1 ) and by Rellich's theorem, the sequence ). This implies in particular that for every ϕ ∈ C 1 c (R N ), that is, u is a weak solution of (P).
We point out that the assumption (f 2 ) is only used in the proof of claim 2.

Regularity of solutions and Pohožaev identity.
We prove in this section that any solution of (P) has some additional regularity; this regularity will be sufficient to establish the Pohožaev identity (1.4). Proposition 3.1 (Improved local regularity of solutions of (P)). If f ∈ C(R; R) satisfies (f 1 ) and u ∈ H 1 (R N ) solves (P), then for every p ≥ 1, u ∈ W 2,p loc (R N ). In particular, proposition 3.1 with the Morrey-Sobolev embeddings imply that a solution u is locally Hölder continuous. If f has additional regularity then regularity of u could be further improved via Schauder estimates.
The assumption (f 1 ) is too weak for the standard bootstrap method as in [8, , then u ∈ L p (R N ) for every p ≥ 1. We extend this result to a class of nonlocal linear equations.

Proposition 3.2 (Improved integrability of solution of a nonlocal critical linear equation). If
Our proof of proposition 3.2 follows the strategy of Brezis and Kato (see also Trudinger [28,Theorem 3]). The adaptation of the argument is complicated by the nonlocal effect of u on the right hand side.
Our main new tool for the proof of proposition 3.2 is the following lemma, which is a nonlocal counterpart of the estimate [5, lemma 2.1]: if V ∈ L ∞ (R N ) + L N 2 (R N ), then for every ǫ > 0, there exists C ǫ such that In the limit α = 0, this result is consistent with (3.2); the parameter θ only plays a role in the nonlocal case.
In order to prove lemma 3.3, we shall use several times the following inequality.
Proof. First observe that ifs > 1,t > 1, satisfy 1 t + 1 s = 1 + α N , the Hardy-Littlewood-Sobolev inequality is applicable and then by Hölder's inequality It can be checked that (3.3) and (3.4) can be satisfied for some µ ∈ R if and only the assumptions of the lemma hold. In particular, 1 t + 1 s = 1 s + 1 t = λ q + 2−λ r = 1 + α N , so that we can conclude.
Taking now, s = t = 2N α , q = r = 2 and λ = 2, we have since |θ − 1| < N −α N , By the Sobolev inequality, we have thus proved that for every u ∈ H 1 (R N ), The conclusion follows by choosing H * and K * such that Proof of proposition 3.2. By Lemma 3.3 with θ = 1, there exists λ > 0 such that for every ϕ ∈ H 1 (R N ), Choose sequences (H k ) k∈N and (K k ) k∈N in L 2N α (R N ) such that |H k | ≤ |H| and |K k | ≤ |K|, and H k → H and K k → K almost everywhere in R N . For each k ∈ N, the form a k : It can be proved that the sequence (u k ) k∈N converges weakly to u in H 1 (R N ) as k → ∞.
For µ > 0, we define the truncation u k,µ : Since |u k,µ | p−2 u k,µ ∈ H 1 (R N ), we can take it as a test function in (3.5): If p < 2N α , by lemma 3.3 with θ = 2 p , there exists C > 0 such that We have thus Since p < 2N α , by the Hardy-Littlewood-Sobolev inequality, In view of the Sobolev estimate, we have proved the inequality lim sup By iterating over p a finite number of times we cover the range p ∈ [2, 2N α ).
and if u ∈ H 1 (R N ) satisfies −∆u + u = (I α * Hu)K, then one has

Regularity of solutions.
Here we prove of the regularity result for the nonlinear problem (P).
Proof of proposition 3.1. Define H : R N → R and K : By the classical bootstrap method for subcritical local problems in bounded domains, we deduce that u ∈ W 2,p loc (R N ) for every p ≥ 1.

Pohožaev identity.
The regularity information that has been gained in the previous sections allows to prove a Pohožaev integral identity. and This proposition implies in particular that if u = 0, then The proof of this proposition is a generalization of the argument for f (s) = s p [24] (see also particular cases [9, lemma 2.1; 22]). The strategy is classical and consists in testing the equation against a suitable cut-off of x · ∇u(x) and integrating by parts [15, proposition 6.2.1; 32, appendix B].

Proof of proposition 3.5. By proposition
can be used as a test function in the equation to obtain The left-hand side can be computed by integration by parts for every λ > 0 as Lebesgue's dominated convergence theorem implies that Similarly, as u ∈ W 2,2 loc (R N ), the gradient term can be written as Lebesgue's dominated convergence again is applicable since ∇u ∈ L 2 (R N ) and we obtain The last term can be rewritten by integration by parts for every λ > 0 as We can thus apply Lebesgue's dominated convergence theorem to conclude that

Solutions and paths.
One of the application of the Pohožaev identity of the previous section is the possibility to associate to any variational solution of (P) a path, following an argument of L. Jeanjean and H. Tanaka [14].
The functionγ is continuous on (0, ∞); for every τ > 0, so thatγ is continuous at 0. By the Pohožaev identity of proposition 3.5, the functional can be computed for every τ > 0 as It can be checked directly that I •γ achieves strict global maximum at 1: for every τ ∈ [0, ∞) \ {1}, I γ(τ ) < I(u). Since lim the path γ can then be defined by a suitable change of variable.

Minimality of the energy and existence of a groundstate.
We now have all the tools available to show that the mountain-pass critical level b defined in (1.6) coincides with the ground state energy level c defined in (1.3), which completes the proof of theorem 1.
Proof of theorem 1. By propositions 2.1 and 2.2, there exists a Pohožaev-Palais-Smale sequence (u n ) n∈N in H 1 (R N ) at the mountain-pass level b > 0, that converges weakly to some u ∈ H 1 (R N ) \ {0} that solves (P). Since lim n→∞ P(u n ) = 0, by the weak convergence of the sequence (u n ) n∈N , the weak lower-semicontinuity of the norm and the Pohožaev identity of proposition 3.5, Since u is a nontrivial solution of (P), we have I(u) ≥ c by definition of the ground state energy level c, and hence c ≤ b.
Let v ∈ H 1 (R N ) \ {0} be another solution of (P) such that I(v) ≤ I(u). If we lift v to a path(proposition 4.1) and recall the definition (1.6) of the mountain-pass level b, we conclude that I(v) ≥ b ≥ I(u). We have thus proved that I(v) = I(u) = b = c, and this concludes the proof of theorem 1.

4.3.
Compactness of the set of groundstates. As a byproduct of the proof of theorem 1, the weak convergence of the translated subsequence of proposition 2.2 can be upgraded into strong convergence. then there exists u ∈ H 1 (R N ) \ {0} such that I ′ (u) = 0 and a sequence (a n ) n∈N of points in R N such that up to a subsequence u n (· − a n ) → u strongly in H 1 (R N ) as n → ∞.
Proof. By proposition 2.2, up to a subsequence and translations, we can assume that the sequence (u n ) n∈N converges weakly to u. Since equality holds in (4.1), and hence (u n ) n∈N converges strongly to u in H 1 (R N ).

Positivity of groundstates.
We now prove that when f is odd, groundstates do not change sign.

Proposition 5.2 (Groundstates do not change sign).
Let f ∈ C(R; R) satisfy (f 1 ). If f is odd and does not change sign on (0, ∞), then any groundstate u ∈ H 1 (R N ) of (P) has constant sign.
Proof. Without loss of generality, we can assume that f ≥ 0 on (0, ∞). By proposition 4.1, there exists an optimal path γ ∈ Γ on which the functional I achieves its maximum at 1/2. Since f is odd, F is even and thus for every v ∈ H 1 (R N ).

Symmetry of groundstates.
In this section, we now prove that groundstates are radial. The argument relies on polarizations. It is intermediate between the argument based on equality cases in polarization inequalities [24] and the argument based on the Euler-Lagrange equation satisfied by polarizations [2,31].
Before proving proposition 5.3, we recall some elements of the theory of polarization [1,6,30,33]. Assume that H ⊂ R N is a closed half-space and that σ H is the reflection with respect to ∂H. The polarization u H : We will use the following standard property of polarizations [6, lemma 5.3].

Lemma 5.5 (Polarization and nonlocal integrals). Let
If F (u H ) = F (u • σ H ), we conclude similarly that u H = u • σ H . Since this holds for arbitrary H, we conclude by lemma 5.6 that u is radial and radially decreasing.

Alternative proof of the existence
In this section we sketch an alternative proof of the existence of a nontrivial solution u ∈ H 1 (R N ) \ {0} such that c ≤ I(u) ≤ b, under the additional symmetry assumption of theorem 2 and in the spirit of the symmetrization arguments of H. Berestycki and P.-L. Lions [3, pp. 325-326]. The advantage of this approach is that it bypasses the concentration compactness argument and delays the Pohožaev identity which is still needed to prove that b ≤ c.
Proof of theorem 1 under the additional assumptions of theorem 2. In addition to (f 1 ), (f 2 ) and (f 3 ), assume that f is an odd function which has constant sign on (0, ∞). With this additional assumption, I • Φ(σ, |v| H ) ≤ I • Φ(σ, v). Therefore, by the symmetric variational principle [29, theorem 3.2], we can prove as in the proof of proposition 2.1 the existence of a sequences (u n ) n∈N and (v n ) n∈N such that as n → ∞, I(u n ) → 0, N −2 (R N ), and v n is radial for every n ∈ N.
As previously, the sequence (u n ) n∈N is bounded in H 1 (R N ); by the Pólya-Szegő inequality, the sequence (v n ) n∈N is also bounded in H 1 (R N ). Since v n is radial for every n ∈ N, the sequence (v n ) n∈N is compact in L p (R N ) for every p ∈ (2, 2N N −2 ) [32, Corollary 1.26]. As u n − v n → 0 in L 2 (R N ) ∩ L 2N N −2 (R N ), the sequence (u n ) n∈N is also compact L p (R N ) for every p ∈ (2, 2N N −2 ). In view of (f 2 ), this implies that F (u n ) → F (u) as n → ∞ in L 2N N +α (R N ) and thus Now one can prove than that u n (· − a n ) converges to a nontrivial solution u ∈ H 1 (R N ) \ {0} as in the proof of proposition 2.2. By (6.1), it also follows that c ≤ I(u) ≤ b.
Finally, employing the Pohožaev identity as in the proof of theorem 1 allows to conclude that c = b.