Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

Abstract: This paper deals with the following Choquard equation with a local nonlinear perturbation: { −∆u + u = ( Iα * |u| α 2 +1 ) |u| α 2 −1u + f (u), x ∈ R2; u ∈ H1(R2), where α ∈ (0, 2), Iα : R2 → R is the Riesz potential and f ∈ C(R,R) is of critical exponential growth in the sense of Trudinger-Moser. The exponent α 2 + 1 is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponent α 2 + 1 and the critical exponential growth of f (u).


Introduction
In the past few years, the following Choquard equation: −∆u + u = Iα * |u| q |u| q− u, x ∈ R N ; u ∈ H (R N ), (1.1) has attracted considerable attention, where N ≥ , α ∈ ( , N), < q < * and Iα : R N → R is the Riesz potential. Physical motivation of (1.1) comes from the case that N = , α = and q = . In this case, Eq.(1.1) is called the Choquard-Pekar equation [21,31], Hartree equation [19] or Schrödinger-Newton equation [27,39], depending on its physical backgrounds and derivations. The existence of a ground state in this case was studied in [21,22,28] via variational arguments. In a pioneering work, Lieb [21] rst obtained the existence and uniqueness of positive solutions to (1.1) with N = , α = and q = . Later, Lions [22,23] got the existence and multiplicity results of normalized solution on the same topic. Moroz and Van Schaftingen [28] proved that (1.1) has a ground state solution if and it has no nontrivial solution when either q ≤ α N + or q ≥ N+α N− . The endpoints of the above interval are critical exponents. The upper critical exponent N+α N− plays a similar role as the Sobolev critical exponent in the local semilinear equations [9,40]. The lower critical exponent α N + comes from the Hardy-Littlewood-Sobolev inequality. So far, there are a variety of interesting results concerning the existence of nontrivial solutions for more general Choquard equation with upper critical growth, see for example, see [6,7,17,24] and the references cited therein. However, to the best of our knowledge, it seems that the only available works regarding the existence of nontrivial solutions for Choquard equation with lower critical exponent and a local nonlinear perturbation are the papers [35,38,41]. In details, Van Schaftingen and Xia [35] proved that the following Choquard equation: admits a ground state solution if there exists Λ > such that f satis es the following three assumptions: (PG) f ∈ C(R, R), f (t) = o(|t|) as t → and f (t) = o(|t| N/(N− ) ) as |t| → ∞. (AR) there exists µ > such that < µF(t) ≤ f (t)t for all t ≠ , where F(t) = t f (s)ds; (ZS) lim inf |t|→ |t| N + > Λ. When N ≥ , by using the Pohoăev identity argument, Wang and Liao [41] obtained the same conclusion only under (PG) and (ZS). When f (u) = λ|u| p− u with λ > and < p < * , Tang, Wei and Chen [38] obtained the existence of a ground state solution to (1.2) for every p ∈ ( , * ). For more existence results on (1.1) or related results, we refer to [1, 4, 5, 8, 12-15, 18, 25, 26, 29, 30, 32-34, 42].
In above-mentioned works [35,38,41], it was only considered the case when f (u) has polynomial growth. When N = , the corresponding Sobolev embedding yields H (R ) ⊂ L s (R ) for all s ∈ [ , +∞), but H (R ) ⊈ L ∞ (R ). In this case, the Pohozaev-Trudinger-Moser inequality in R can be seen as a substitute of the Sobolev inequality, which was rst established by Cao in [11], see also [2,10], and reads as follows. This notion of criticality was introduced by Adimurthi and Yadava [3], see also de Figueiredo, Miyagaki and Ruf [16], for the study of the planar Schrödinger equation which is the maximal growth that allows to treat the problem variationally in H (R ). Inspired by [16,35,38,41], in the present paper, we consider the following planar Choquard equation: where α ∈ ( , ), f satis es (F1) or (F1 ), and Iα : R → R is the Riesz potential de ned by In (1. 7), as what mentioned before, α + is the lower critical exponent coming from the Hardy-Littlewood-Sobolev inequality. Naturally, we are interested in whether (1.7) admits a nontrivial solution or a ground state solution when f has critical exponential growth in the sense of Trudinger-Moser. The double di culties due to both the lower critical exponent α + and the critical exponential growth of f enforce the implementation of new ideas and tricks.
To state our theorems, we need to introduce some notations and assumptions. By the Hardy-Littlewood-Sobolev inequality, one has In view of [20,Theorem 4.3], the sharp constant S is achieved by a function u ∈ H (R ) if and only if for every for a ∈ R , A > and γ > . In what follows, we let A = A and A is determined by (1.10) . Let t be the unique positive root in the interval ( , S α ) of the following equation and let s be the unique positive root in the interval ( , S α ) of the following equation where a+ = max{a, }.
In addition (F1) and (F1 ), we also assume f satis es the following conditions: (F5) there exist p > and λ > λ such that (WN) f (t) |t| is non-decreasing on (−∞, ) ∪ ( , ∞). In view of (1.8) and Lemma 1.1, under (F1) (or (F1 )) and (F2), the energy functional I : H (R ) → R associated with (1.7) is continuously di erentiable, and (1.14) moreover its critical points correspond to the weak solutions of (1.7). As usual, a solution is called a ground state solution if its energy is minimal among all nontrivial solutions. Our main results are as follows.
is the Nehari manifold of I.
Then (1.7) has a solutionū ∈ N such that The paper is organized as follows. In Section 2, we give some useful lemmas. We give the proofs of Theorems 1.2 and 1.3 in Sections 3 and prove Theorems 1.4 and 1.5 in Section 4.
Throughout the paper we also make use of the following notations: • H (R ) denotes the usual Sobolev space equipped with the inner product and norm • L s (R )( ≤ s < ∞) denotes the Lebesgue space with the norm u s = R |u| s dx /s .

Some useful lemmas
By a simple calculation, we can verify the following lemma. Lemma 2.1. Assume that (F1) (or (F1 )), (F2) and (WN) hold. Then the following two inequalities hold: Inspired by [37], we establish a key functional inequality as follows.
From Lemma 2.2, we have the following corollary.
By Lemma 2.4 and the classical mountain pass theorem [40], we can prove the following lemma by a standard argument. I(γ(t)) By Lemma 2.4, one has ζ ( ) = and ζ (t) > for t > small and ζ (t) < for t large. Therefore max t∈( ,∞) ζ (t) is achieved at some tu > so that ζ (tu) = and tu u ∈ N.
From Corollary 2.3 and Lemma 2.6, we have N ≠ ∅ and the following minimax characterization. Next, we apply the non-Nehari manifold method introduced in [36] to prove the following lemma which is key to obtain a Nehari-type ground state solution.
From (WN) and (2.4), we have By virtue of Corollary 2.4, one can get that Hence, by (2.15), (2.17) and (2.18), one has Now, we can choose a sequence {n k } ⊂ N such that Let u k = u k,n k , k ∈ N. Then, going if necessary to a subsequence, we have Next, we give an estimate on the energy level m, which is essential in ensuring compactness.
Let t * = S α . Then we can choose ϵ > such that We set By (2.21), we can choose ε ∈ ( , ) such that Now we de ne a function φε(t) as follows: The above four subcases and Lemmas 2.5 and 2.7 show that max{c , m} ≤ max t≥ I(tUε) < m * .
Proof. As in the proof of Lemma 2.10, we set U(x) = A ( + |x| ) − . By a simple calculation, we have Now we de ne a function φ(t) as follows: The above three subcases and Lemmas 2.5 and 2.7 show that max{c , m} ≤ max t≥ I(tUε) < ω .

Sub-critical case
Then (3.1) and Lemma 1.1 ii) lead to which is a contradiction due the fact that c * ≤ m < m * . This shows that δ > . The rest of the proof is standard, so we omit it. Proof.