Existence of groundstates for a class of nonlinear Choquard equations in the plane

We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equation $$ -\Delta u+u=(I_\alpha*F(u))F'(u)\qquad\text{in }\mathbb{R}^2, $$ where $I_\alpha$ is the Riesz potential of order $\alpha$ on the plane $\mathbb{R}^2$ under general nontriviality, growth and subcriticality on the nonlinearity $F \in C^1 (\mathbb{R},\mathbb{R})$.


Introduction
We are interested in the existence of nontrivial solutions to the class of nonlinear Choquard equations of the form where N ∈ N = {1, 2, . . . }, ∆ is the standard Laplacian operator on the Euclidean space R N , I α : R N → R is the Riesz potential of order α ∈ (0, N ) defined for each x ∈ R N \ {0} by and a nonlinearity is described by the function F ∈ C 1 (R, R). Solutions of the equation (P) are at least formally critical points of the energy functional defined for a function u : R N → R by (1) In the particular case where for each s ∈ R, F (s) = s 2 /2, solutions to the Choquard equation (P) are standing waves solutions of the Hartree equation. In particular when N = 3 and α = 2, the problem (P) has arisen in various fields of physics: quantum mechanics [20], one-component plasma [11] and self-gravitating matter [15]. In these cases, many existence results have been obtained in literature, with both variational [11,13,14] and ordinary differential equations techniques [6,15,21] (see also the review [18]). Such methods extend also to the case of homogeneous nonlinearities [16].
Date: December 7, 2016. 2010 Mathematics Subject Classification. 35J91 (35J20) . This work was supported by the Projet de Recherche (Fonds de la Recherche Scientifique-FNRS) T.1110.14 "Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations".
When the nonlinearity F is not any more homogeneous, it has been shown that the Choquard equation (P) has a nontrivial solution if the nonlinearity F satisfies the following hypotheses [17]: for every s > 0; (F ′ 2 ) lim s→0 F (s)/|s| 1+ α N = 0 = lim s→0 F (s)/|s| N+α N−2 . The solution u is a groundstate, in the sense that u minimizes the value of the functional I among all nontrivial solutions. The assumptions (F ′ 0 ), (F ′ 1 ) and (F ′ 2 ) are rather mild and reasonable and are "almost necessary" in the sense of Berestycki and Lions [3]: the nontriviality of the nonlinearity condition (F ′ 0 ) is clearly necessary to have a nontrivial solution; the assumption (F ′ 1 ) secures a proper variational formulation of the problem (P) by ensuring that the energy functional I is well-defined on the natural Sobolev space H 1 (R N ) through the Hardy-Littlewood-Sobolev and Sobolev inequalities; the condition (F ′ 2 ) is a sort of subcriticality condition with respect to the limiting-case embeddings. The analysis by a Pohožaev identity shows that the assumptions (F ′ 1 ) and (F ′ 2 ) are necessary in the homogeneous case F (s) = s p /p [16].
The results in [17] can thus be seen as a counterpart for Choquard-type equations of the result of Berestycki and Lions [3] which give similar "almost necessary" conditions for the existence of a groundstate to the equation The latter equation can be at least formally be obtained by (P) by passing to the limit as α → 0 and setting G = F 2 /2. Whereas the above-mentioned almost necessary conditions for existence of the Choquard equation (P) and for the scalar field equation (2) have been obtained in higher dimensions N ≥ 3, the latter result has been extended to the two-dimensional case [4], under the following assumptions This raises naturally the question whether there is a similar existence result for the Choquard equation (P) in the planar case.
In the present work, we provide a general existence result for groundstate solutions of problem (P) in the planar case N = 2, which is a two-dimensional counterpart of [17] and a counterpart for the Choquard equation of [4]. The counterparts of (F ′ 0 ), (F ′ 1 ), (F ′ 2 ) we need are the following: Our main result reads as follows: Theorem 1.1. If N = 2 and F ∈ C 1 (R, R) satisfies the conditions (F 0 ), (F 1 ) and (F 2 ), then the problem (P) has a groundstate solution u ∈ H 1 (R 2 ) \ {0}, namely the function u solves (P) and Let us discuss the assumptions of Theorem 1.1. As above, the assumption (F 0 ) is necessary for the existence of a nontrivial solution. As before, the condition (F 1 ) ensures needed the well-defineteness of the energy functional on the whole space H 1 (R 2 ). It has a different shape, because in dimension N = 2, the critical nonlinearity for Sobolev embeddings is not anymore a power but rather an exponential-type nonlinearity. More precisely, the integral of min{1, u 2 }e θ|u| 2 on R 2 is uniformly controlled on H 1 0 (B 1 ) if and only if θ´B 1 |∇u| 2 ≤ 4π (see [1,19]); this is why the parameter θ > 0 appears in condition (F 1 ). It will appear that the condition (F 1 ) is strong enough at infinity. Indeed, by integrating the function F ′ , it is possible to observe that for every θ > 0, A subcriticality condition still needs to be imposed around 0; that is the goal of the subcriticality condition (F 2 ). The assumptions (F 0 ), (F 1 ) and (F 2 ) are still almost necessary: in the case F (s) = s p p , they are satisfied if and only if p > 1 + α 2 , and for p ≤ 1 + α 2 the Choquard equation (P) has no nontrivial solutions (see [16]).
In order to prove Theorem 1.1 the constraint minimization technique used in [3,4] for the local problem (2) does not seem to work, as it introduces a Lagrange multiplier that cannot be absorbed through a suitable dilation because of the presence of three different scalings in the equation and of the nonhomogeneity of the nonlinearity.
Following [17], we use a mountain-pass construction. We start by constructing a Palais-Smale sequence for the mountain-pass level To avoid relying on an Ambrosetti-Rabinowitz superlinearity condition, we use a scaling trick due to Jeanjean [9], which allows to construct Pohožaev-Palais-Smale sequence (Proposition 3.1), namely a Palais-Smale sequence which, in addition, satisfies asymptotically the Pohožaev identity Such a condition will imply quite directly the boundedness of the sequence in the space H 1 (R 2 ) and it will be crucial to get the convergence, hence the existence of a solution (Proposition 4.1).
We are left with showing that the solution u is actually a groundstate. To prove this, we first show that the solution u itself satisfies the Pohožaev identity (Proposition 5.2). This will follow by simple calculations once a suitable regularity result is established (Proposition 5.1); this regularity turns out to be easier to prove from the assumption (F 1 ) than in the higher-dimensional case [17] where a suitable nonlocal Brezis-Kato regularity had to be proved. The last ingredient that we need is an optimal path γ v ∈ Γ associated to any solution v of (P). The construction of such paths (Proposition 5.3) is inspired by [10,17] but it is more delicate in our two-dimensional case than in the higher dimensions N ≥ 3, because dilations t → v(·/t) ∈ H 1 (R N ) are not anymore continuous at t = 0 when N = 2.
The content of the paper is the following: in Section 2 we provide some technical preliminaries; in Section 3 we construct the Pohožaev-Palais-Smale sequence; in Section 4 we show that the sequence converges to a solution of (P); in Section 5 we prove that u is actually a groundstate. In the last section we also state some qualitative result concerning the solutions, which can be proved directly following [17].

Preliminaries
In this section we present some preliminary results which we will need throughout the rest of this paper. We start by reformulating in a more convenient form the Moser-Trudinger inequality of Adachi and Tanaka [1]. This quantitative estimate will play a crucial role throughout the paper. together with the elementary inequalities valid for every s ≥ 0,

Proposition 2.1 (Moser-Trudinger inequality). For any
We will also use the Hardy-Littlewood-Sobolev inequality to deal with the nonlocal term (see for example [12,Theorem 4.3]): Combining the last two results with the assumption on F and (3) we deduce that the energy functional is well-defined on H 1 (R 2 ): , then the energy functional I defined by (1) is well-defined and continuously differentiable.
Proof. We first consider the superposition map E defined for each u ∈ H 1 (R 2 ) and x ∈ R 2 by E(u)(x) = F ′ (u(x)). We claim that E is well-defined and continuous as a map from H 1 (R 2 ) to L 4/α (R 2 ). Indeed by assumption (F 1 ), for every θ > 0, and s ∈ R, we have on R 2 , where the right-hand side is integrable in view of the Moser-Trudinger inequality (Proposition 2.1); therefore the map E : If now the sequence (u n ) n∈N converges to u in H 1 (R 2 ), then we can assume without loss of generality that ν := sup n∈N´R 2 |∇u| 2 < απ 2θ and that (u n ) n∈N converges to u almost everywhere. We have then for some constant C ≥ 0, If we consider the set A λ n = {x ∈ R 2 | |u n (x)| ≥ λ}, we have by Lebesgue's dominated convergence theorem, for every λ > 0, On the other hand, we have by the Cauchy-Schwarz inequality, the Chebyshev inequality and the Moser-Trudinger inequality (Proposition 2.1) This allows to conclude that the map E : We now consider the map F : and thus for almost every x ∈ R 2 , It follows thus from the first part of the proof that F is well-defined from For the differentiability we consider a sequence (u n ) n∈N converging strongly to u in H 1 (R 2 ). We observe that for each n ∈ N, and thus by Hölder's inequality By the convergence of the sequence (u n ) n∈N and the continuity of the functional E, it follows that, as n → ∞, that is, E represents the Fréchet differential of the functional F. Since E is continuous, it follows that F is of class C 1 .
Finally, we consider the quadratic form Q defined for f ∈ L 4/(2+α) by By the Hardy-Littlewood-Sobolev inequality (Proposition 2.2), the quadratic form Q is bounded on bounded sets of the space L 4/(2+α) (R 2 ). This implies that Q is continuously differentiable and thus the functional is continuously differentiable. By the smoothness of the norm on a Hilbert space, we conclude that the functional I is continuously differentiable.
Finally, we will use the following improvement of Proposition 2.2 when one has some more L p integrability: Proof. The result is classical. We give its short proof for the convenience of the reader. By choosing p, q in those range we have (2 − α) q q−1 < 2 < (2 − α) p p−1 ; therefore, through splitting the integral and Hölder inequality we get for every x ∈ R 2

Construction of a Pohožaev-Palais-Smale sequence
In this section we show the existence of a Pohožaev-Palais-Smale sequence at the level b defined by (4). In other words, we construct a sequence of almost critical points which asymptotically satisfies the equation (P) and the Pohožaev identity (6).
To prove Proposition 3.1, we first need to show that the energy functional I has the mountain pass geometry, namely that the mountain pass level b is well-defined and nontrivial: Proof. We start by showing the finiteness of b, which will be done as in [17,Proposition 2.1]. By the definition of the set b, it is sufficient to show that Γ = ∅, which in turn is equivalent to find u 0 ∈ H 1 (R 2 ) such that I(u 0 ) < 0. By the assumption (F 0 ), we can take s 0 such that F (s 0 ) = 0 and we find therefore, for some t 0 ≫ 0, the function u 0 := v t 0 satisfies I(u 0 ) < 0.
Let us now show that b > 0. By the definition of b, it is equivalent to show that there exists ε > 0 such that for every path γ ∈ Γ there exists t γ ∈ [0, 1] with I(γ(t γ )) ≥ ε > 0.

Convergence of the Pohožaev-Palais-Smale sequence
In this Section we will construct a nontrivial solution of (P) from the sequence given by Proposition 3.1.

Proposition 4.1.
If the function F ∈ C 1 (R, R) satisfies (F 1 ) and (F 2 ) and the sequence (c) P(u n ) → n→∞ 0; then, up to subsequences, one of the following occurs: We follow the strategy of [17, Proposition 2.2]. Since the gradient does not appear in the Pohožaev identity (6), it will be more delicate to show that the nonlocal term does not vanish.
Proof of Proposition 4.1. We assume that the first alternative does not hold, namely lim inf By writing for each n ∈ N
We now want to prove that u n does not vanish. We will use the following inequality [13, Lemma I.1] (see also [16, and we will show that the right-hand side term is bounded from below by a positive constant, for every p > 2. By the assumption (F 2 ) and (3), for every ε > 0 there exists C ε,θ > 0 such that The quantity ε being arbitrary, we get´B 1 (xn) |u n | p ≥ 1 C for some x n ∈ R 2 , for n large enough.
We can now consider the translated sequence (u n (· − x n )) n∈N . Since the problem (P) is invariant by translation, this sequence will satisfy the hypotheses of the present proposition, hence we will still denote it as (u n ) n∈N and we will assume that x n = 0 for all n ∈ N. Since lim inf n→∞´B 1 |u n | p > 0, we can assume that this sequence (u n ) n∈N converges weakly to u ∈ H 1 (R 2 ) \ {0}. We just have to show that u solves (P).
Proof. By Proposition 3.1, I admits a Pohožaev-Palais-Smale sequence (u n ) n∈N at the level b. We apply Proposition 4.1 to (u n ) n∈N . If the first alternative occurred, then we would have I(u n ) → I(0) = 0 as n → ∞, in contradiction with Lemma 3.2. Therefore, the second alternative must occur, and in particular we get a solution u ∈ H 1 (R 2 ) \ {0} of (P).

From solutions to groundstates
We start by providing a local regularity result for solution of (P). This result can be obtained quite directly because our growth assumption (F 1 ) gives a good control on I α * F (u) which, in turn, permits to apply a standard bootstrap method. The equivalent result in higher dimension N ≥ 3 is more delicate to prove (see [17,Theorem 2]) because of the relative weakness of assumption (F ′ 1 ). Proposition 5.1. If F ∈ C 1 (R, R) satisfies the condition (F 1 ) and if the function u ∈ H 1 (R 2 ) solves the problem (P), then u ∈ W 2,p loc (R 2 ) for every p ≥ 1. Proof. By (3) and Lemma 2.1 we deduce that if v ∈ H 1 (R 2 ) then F (v) ∈ L p (R 2 ) for every p ≥ 4 2+α . Since 2 α > 4 2+α , by Proposition 2.4 inequality we get I α * F (v) ∈ L ∞ (R 2 ). Therefore, any solution u of (P) verifies with F ′ (u) ∈ L p loc (R 2 ) for every p ≥ 1 because of (F 1 ). By standard (interior) regularity theory on bounded domains (see for example [7,Chapter 9]) we deduce that u ∈ W 2,p loc (R 2 ). The extra regularity just proved allows to prove that solutions of (P) satisfy the Pohožaev identity (6). The proof of the Pohožaev identity is classical and it is based on testing (P) against a suitable cut-off of x · ∇u(x), therefore it will be skipped. Details can be found in [17,Theorem 3].

Proposition 5.2 (Pohožaev identity).
If F ∈ C 1 (R, R) satisfies (F 1 ) and u ∈ H 1 (R 2 ) ∩ W 2,2 loc (R 2 ) solves (P), then The Pohožaev identity allows us to show that the mountain pass solution is actually a groundstate. We will argue like [10, Lemma 2.1; 17, Proposition 2.1], associating to any solution v a path γ v ∈ Γ passing through v. The main difficulty here is that the integral of |∇u| 2 is invariant by dilation, therefore we are not allow to join v with 0 by just taking dilations t → v · t . To overcome this difficulty, we will combine properly dilatations and multiplication by constants [10].
We are now in position to prove the main theorem of this work.
Proof of Theorem 1.1. Let (u n ) n∈N be the Pohožaev-Palais-Smale sequence given by Proposition 3.1. Then, by Proposition 4.1, it converges weakly to a solution u ∈ H 1 (R 2 )\ {0} of (P). By definition of groundstate, I(u) ≥ c and, by Proposition 5.2, we have P(u) = 0 (Proposition 5.2 is applicable in view of Proposition 5.1). Arguing as in [17, Theorem 1], we get successively If v ∈ H 1 (R 2 ) \ {0} is another solution of the Choquard equation (P), we apply Proposition 5.3 to v: The solution v being arbitrary, by definition of groundstate one has b ≤ c. Putting everything together, we get c ≤ I(u) ≤ b ≤ c, The proof is complete.
We point out as a corollary of the proof of Theorem 1.1, that the convergence in Proposition 4.1 turns out to be actually a strong convergence in H 1 (R 2 ) and that this gives as a byproduct a compactness property of the set of groundstates of (P). Then, there exists u ∈ H 1 (R 2 ) \ {0} solving (P) and a sequence (x n ) n∈N in R 2 such that, up to subsequences, u n (· − x n ) → n→∞ u strongly in H 1 (R 2 ). Moreover, the set of groundstates S c := u ∈ H 1 (R 2 ); u solves (P) and I(u) = c is compact, up to translations, in H 1 (R 2 ).
Proof. We apply Proposition 4.1; the first alternative is excluded by our assumption and the continuity of the functional I at 0. Therefore we get, up to translations, u n ⇀ u as n → ∞ in H 1 (R 2 ) and the function u ∈ H 1 (R 2 ) \ {0} solves (P). As in the proof of Theorem 1.1, we get (13) lim inf from which it follows that u n → u strongly in H 1 (R 2 ) as n → ∞.
To show the compactness of the set of groundstates S c , we consider an arbitrary sequence (u n ) n∈N in S c . Because of Proposition 5.2, it verifies P(u n ) = 0 for every n ∈ N, so it satisfies the hypotheses of Proposition 4.1 and of the first part of the present corollary; therefore, up to subsequences and translations it will converge to some u which solves (P) and, by the continuity of the functional I in H 1 (R 2 ), we get u ∈ S c . We conclude this paper by the following result on additional qualitative properties of the solution u. Proposition 5.5. If F is even and nondecreasing on (0, ∞) and u is a groundstate solution of (P), then u has constant sign and is radially symmetric with respect to some point a ∈ R N .
Proof. The proof is the same as [17, Propositions 5.2 and 5.3]. We briefly sketch the argument for the convenience of the reader.
To prove the constant-sign property, consider the path γ u defined in Proposition 5.3. Since F is an even function, I(|v|) = I(v) for every v ∈ H 1 (R 2 ), hence I(|γ u (t)|) < I (|γ u (1/2)|) = b for every t ∈ [0, 1] \ {1/2}. From this, one easily deduces that the function |u| is a groundstate solution of (P); since F ′ ≥ 0, we can apply the strong maximum principle and get |u| > 0, namely u has constant sign. Without loss of generality we assume now that u ≥ 0.
For the symmetry, we follow the strategy of Bartsch, Weth and Willem [2] and its adaptation to the Choquard equation [16,17]. For any closed half space H ⊂ R 2 we consider the reflection σ H with respect to H and define, for every u ∈ H 1 (R 2 ), the polarization (see for example [5]) We first observe that [