Normalized solutions for a Choquard equation with exponential growth in R 2

. In this paper, we study the existence of normalized solutions to the following nonlinear Choquard equation with exponential growth


Introduction
This paper concerns the existence of normalized solutions to the following nonlinear Choquard equation with exponential growth where a > 0 is prescribed, λ ∈ R, α ∈ (0, 2), I α denotes the Riesz potential defined by where Γ represents the gamma function, * indicates the convolution operator, F (t) = t 0 f (τ )dτ .The nonlinearity f satisfies some suitable conditions that will be specified later.
Equation (1.1) arises from seeking the standing wave solutions of prescribed mass to the following time-dependent nonlinear Choquard equation where f (e iλt z) = e iλt f (z) and F (e iλt z) = t 0 f (e iλt z)dz = e iλt F (z) for any λ, t, z ∈ R. As we all know, an important feature of equation (1.2) is conservation of mass with the L 2 -norms |Φ(•, t)| 2 of solutions are independent of t ∈ R. To find stationary states, one makes the ansatz Φ(x, t) = e iλt u(x), which still ensure conservation of mass with the L 2 -norms.
We mention that Schrodinger equations with prescribed L 2 -norms has received extensive attention in recent years.Jeanjean [11] considered the existence of solutions of prescribed mass to the following Schrödinger equations where a > 0 is prescribed, N ≥ 1 and f satisfies some subcritical growth conditions.Using the minimax approach and a compactness argument, the author obtained the existence of normalized solutions for equations (1.3).In [13], Jeanjean and Lu complemented and generalized the known results in [11].In [23], Soave studied the normalized solutions for equations (1.3) with combined nonlinearities, where f (t) = µ|t| q−2 t + |t| p−2 t and 2 < q ≤ 2 + 4 N ≤ p < 2 * .For f has a critical growth in the Sobolev sense, Soave [24] obtained the existence of ground states and raised some questions with combined nonlinearities.For more results in normalized solutions for the equations (1.3), the reader may refer to [6,7,12,15,26,4,1] and references therein.
On the other hand, we mention that the study of our problem is based on some interesting results of the following Choquard equation This nonlocal equation plays an important role in quantum theory and description of finite range multi-body interaction.For N = 3, α = 2 and f (s) = s, Fröhlich [10] introduced this equation to study the modeling of quantum polaron.Moroz and Van Schaftingen [18,19] obtained the existence of a ground state solution for equation (1.4) and studied its some properties.For N = 2 and f has exponential critical growth, equation (1.4) has been investigated by some authors; see e.g.[2,8,22] and the references therein.For more classical results regarding Choquard equation, we refer to [20] for a good survey.When the L 2 -norms |u| 2 is prescribed, equation (1.4) has important physical significance: in Bose-Einstein condensates and the nonlinear optics framework.In [14], Li and Ye firstly considered the following Choquard equation where λ cannot be prescribed but appears as a Lagrange multiplier in a variational approach.The authors in [14] used a minimax procedure and the concentration compactness to show equation (1.5) has at least a weak solution.For N = 3, Yuan et al. [30] complemented and generalized the known results in [14].In [5], Bartsch et al. proved the existence of a least energy solution of equation (1.5) in all dimensions N ≥ 1, which is simpler and more transparent than the one from [14].For critical case, Ye et al. [29] obtained the existence of ground states for the critical Hartree equation with perturbation.
For more results regarding equation (1.5), the reader may refer to [16,28] and references therein.We know that classical Sobolev embedding that H 1 (R N ) is continuously embedded in L q (R N ) for all q ∈ [2, 2 * ], where 2 * = 2N/(N − 2).Thus, we know that 2 * = ∞, if N = 2.In this case, Ji et al. [3] firstly studied equation (1.5) with exponential critical nonlinearities.Using the minimax approach, the authors firstly obtained results for normalized problem with two-dimensional exponential critical growth.In addition, when N ≥ 3, the authors in [3] complemented some recent results found in [24] is nonincreasing in (0, ∞) and nondecreasing in (0, ∞).
In the subcritical case, we assume that f satisfies the following condition (f 6 ) f has exponential subcritical growth, i.e. for all γ > 0 we have The first result of this paper can be stated as follows: Theorem 1.1.Assume that f satisfies (f 1 )-(f 3 ) and (f 6 ).Then equation (1.1) has a weak solution (λ, u) with λ > 0 and u ∈ H 1 (R 2 ).Moreover, if (f 5 ) is also assumed, then u can be chosen as a nontrivial ground state solution of equation (1.1).
Motivated by the research made in the Choquard equations and [3], in this paper we consider the exponential critical growth for Choquard equations in R 2 .We recall that in R 2 , the natural growth restriction on the function f is given by the Trudinger-Moser inequality.
Remark 1.2.In this article, we suppose that f satisfies (f 6 ) or (f 7 ), which means f has subcritical exponential growth or critical exponential growth.This notion of criticality is motivated by the inequality of Trudinger and Moser; see [21,25].
The paper is organized as follows: In Section 2, the variational setting and some preliminary results are presented.In Section 3, we introduce the geometry structure related to equation (1.1).In Section 4, we give some properties about the (P S) sequence.In Section 5, we complete the proof of subcritical case.Section 6 is devoted to give an estimate for the minimax level of the critical case.Finally in Section 7, we complete the proof of critical case.
Notation: From now on in this paper, otherwise mentioned, we use the following notations:

•
denotes the usual norm of the Sobolev space H 1 (R 2 ).

Preliminaries and functional setting
In this section, we give some preliminary results and outline the variational framework for (1.1).
and γ < 4π, then there exists a constant C(M, γ), which depends only on M and α, such that Now we recall the Hardy-Littlewood-Sobolev inequality, see [17].
This implies that we must require In order to apply variational methods, we recall that H 1 (R 2 ) denotes the usual Sobolev space with the inner product and norm Solutions of equation (1.1) correspond to critical points of the energy functional J : and constrained to the L 2 -torus The parameters λ ∈ R will appear as Lagrange multiplier.By Proposition 2.1 and 2.2, we know that (I α * F (u))F (u) ∈ L 1 (R 2 ), which implies that J is well defined.It is easy to see that J is of class C 1 , and that it is unbounded from below on T (a).It is well known that critical points of J will not satisfy the Palais-Smale condition, as a consequence we recall that solutions of (1.1) satisfy the Pohozaev identity Now, we introduce the L 2 -invariant scaling s ⋆ u(x) := e s u(e s x).To be more precise, for u ∈ T (a) and s ∈ R, let H : A straightforward calculation shows that J′ u (s) = P (H(u, s)).In order to overcome the loss of compactness of the Sobolev embedding in whole R 2 , in our opinion, we work on the space H 1 rad (R 2 ) to get some compactness results.Thus, we define From (f 1 )-(f 3 ) and if f (t) has subcritical exponential growth at +∞, we have the following immediate result: fix q > 2 and τ > 3, for any ε > 0 and γ > 0, there exists a constant κ ε > 0, which depends on q, γ, ε, such that and Similarly, if f (t) has critical exponential growth at +∞ with critical exponent γ 0 , then fix q > 2 and τ > 3, for any ε > 0 and γ > γ 0 , there exists a constant κ ε > 0, which depends on q, γ, ε and µ, such that and In Section 3 and 4, we consider equation (1.1) in subcritical case or critical case.By (f 6 ), we know that γ u 2 < (2 + α)π for γ > 0 close to 0 and u is bounded in H 1 (R 2 ).Similarly, by (f 7 ), we know that γ u 2 < (2 + α)π for γ > γ 0 close to γ 0 and u 2 < (2+α)π γ 0 .Hence, in what follows, we consider the problem for a unified condition γ u 2 < (2 + α)π.

The minimax approach
To find a solution of (1.1), in this section, we show that J on T r (a)×R possesses a kind of mountain-pass geometrical structure.Proof.By a straightforward calculation, it follows that and From the above equalities, fixing ξ > 2, we have By (2.4) and (2.6), we obtain where q > 2. Hence, for all γ H(u, s) 2 < (2 + α)π.Hence, using Proposition 2.2 and Hölder equality, we deduce that Then, there exists t > 1 close to 1 such that Then, by (3.4), we have where t ′ = t t−1 , and t > 1 close to 1. Now, by using (3.2), we obtain Thus, by τ − 1 > 0 and qt ′ − 2 > 0, we know that In order to show (ii), we define Then, note that by (3.1), Set where t = e s .By (f 3 ) we know Therefore, we obtain Since 2θ − (2 + α) > 2, the last inequality yields J(H(w, s)) → −∞ as s → +∞.
As a byproduct of the last lemma is the following corollary.
, there exists K(a) > 0 small enough such that J(u) > 0.Moreover, Proof.Arguing as in the last lemma, we have for all w ∈ T r (a) In what follows, we fix u 0 ∈ T r (a) and apply Lemma 3.1, 3.2 and Corollary 3.1 to get two numbers s 1 < 0 and s 2 > 0, of such way that the functions u 1 = H(u 0 , s 1 ) and u 2 = H(u 0 , s 2 ) satisfy > 2K(a), J(u 1 ) > 0 and J(u 2 ) < 0. Now, following the ideas from Jeanjean [11], we fix the following mountain pass level given by where Then, we obtain that m(a) ≥ J * > 0.

Palais-Smale sequence
In this section, we take {u n } ⊂ T r (a) demotes the (P S) sequence associated with the level m(a) for J. Using ûn = H(u n , s n ), we know that {û n } is a (P S) sequence associated with the level m(a) for J. Thus, we have J(u n ) → m(a), as n → +∞, (4.1) for some sequence {λ n } ⊂ R, and is achieved.Then, in order to prove it, we give some properties of the (P S) sequence.
Proof.Combining (4.1) and ( 4.3), we obtain By (f 3 ) we have Thus we obtain On the other hand, using again (4.1), it follows that Taking (4.4) in (4.5), we deduce that Then, the (P S) sequence By Proposition 2.1, similar to [3, Lemma 4.1], we have the following Lemma.
) and u n (x) → u(x) a.e. in R, then Proof.Setting Therefore, {h n } is a bounded sequence in L t (R 2 ).Thus, for some subequence of {u n }, still denoted by itself, we obtain that Now, we show that where t ′ = t t−1 .Then, by the embedding ) is compact, we have Hence, we get (4.12).Together (4.11) with (4.12), we know Then, the proof is complete.
By Lemma 4.3, we have the following two important Corollaries.
Proof.By [2, Lemma 4.1], we know Hence, for any φ ∈ C ∞ 0 (R 2 ), we have Proof.By (4.2), we know that which together with the sequence {u n } is bounded in H 1 (R 2 ) imply that {λ n } is a bounded sequence.Thus, by |u n | 2 2 = a 2 , we have The equality (4.16) together with the limit (4.3) lead to lim sup which completes the proof.
Proof.For u ∈ T r (a) and s ∈ R, we know (I α * F (e s u))F (e s u)dx.
Then, we have J′ where By Lemmas 3.1 and 3.2, we know that there exists at least a s 0 ∈ R such that f ′ u (s) |s=s 0 = 0.For any t ∈ R, t = 0, from (f 3 ) and (f 5 ), we see that F (st) s 2+ α 2 is strictly increasing in s ∈ (0, ∞) and F (st) s 2+ α 2 is nondecreasing in s ∈ (0, ∞).This implies ψ(s) is strictly increasing in s ∈ (0, ∞) and there is at most one s(u) ∈ R such that H(u, s(u)) ∈ P(a).Thus, for u ∈ T r (a), there exists a unique maximum at a point s(u) ∈ R such that H(u, s(u)) ∈ P(a).

From this
→ 0, which is absurd, because m(a) > 0.Then, we show that λ > 0. By Lemma 4.4 and (f 3 ), there exists a bounded sequence {λ n } such that lim sup From this for some subsequence, still denoted by {λ n }, we can assume that λ n → λ > 0 as n → ∞.
Therefore λ > 0.Then, by Corollary 4.1, the equality (4.2) implies that Thus, we deduce that P (u) = 0. Now, we obtain that u n ⇀ u = 0. Then we show the strong convergence that u n → u in H 1 rad (R 2 ).The proof is divided into two steps.

On the mini-max level
In this section, we obtain an upper bound for the minimax level.Thus, we obtain an upper bound for u 2 , which is important for exponential critical problem.7 Proof of Theorem 1.2 In this section, we assume that f has critical growth and restrict our study in H 1 rad (R 2 ).Proof of Theorem 1.2.By Lemma 6.Following a similar argument as Section 5, we complete the proof of Theorem 1.2.