Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term

In this paper, we investigate the following a class of Choquard equation \begin{document}$\begin{equation*} -Δ u+u = (I_α*F(u))f(u) \ \ \ \ \ \ {\rm in} \ \mathbb{R}^N,\end{equation*}$ \end{document} where \begin{document}$N≥ 3,~α∈ (0,N),~I_α$\end{document} is the Riesz potential and \begin{document}$F(s) = ∈t_{0}^{s}f(t)dt$\end{document} . If \begin{document}$f$\end{document} satisfies almost necessary the upper critical growth conditions in the spirit of Berestycki and Lions, we obtain the existence of positive radial ground state solution by using the Pohožaev manifold and the compactness lemma of Strauss.

1. Introduction and main result. In recent years, many authors considered the following Choquard equation −∆u + u = (I α * F (u)) f (u), u ∈ H 1 (R N ), (1) where N ≥ 3, I α is the Riesz potential of order α ∈ (0, N ) defined for every x ∈ R N \ {0} by I α = Γ( N −2 2 ) 2 α π N/2 Γ( α 2 )|x| N −α , F (s) = s 0 f (t)dt and f ∈ C(R, R). The Choquard equation (1) appeared in the contexts of various physical models. Especially, when N = 3, α = 2, F (u) = |u| 2 and f (u) = u in equation (1), that is, which was used to describe the quantum theory of a polaron at rest by Pekar [22] in 1954. Later, equation (2) was proposed to describe as a certain approximation to Hartree-Fock theory of one-component plasma in the modeling of an electron trapped in its own hole by Choquard [12] in 1976. In the 1990s, the same equation reemerged as a model of self-gravitating matter [5,10,21,23] and was known in that context as the Schrödinger-Newton equation.
For the case that F (s) = |s| p with N +α N ≤ p ≤ N +α N −2 in problem (1), there exist lots of papers in which the authors considered the existence and qualitative properties of solutions, where the exponent p = N +α N −2 (or α N + 1) is called the upper (or lower) critical exponent with respect to the Hardy-Littlewood-Sobolev inequality (see [14]), which we sketch here for the readers' convenience. Assume that k ∈ L s (R N ) and h ∈ L t (R N ). Then one has in which 1 < s, t < +∞, 0 < α < N and 1 s + 1 t + α N = 2. Further results for related problems can be found in [11,13,16,17,19,20,25] and references therein.
While many researchers considered the autonomous Choquard equation involving general nonlinear term for recent years. This situation contrasts with the striking existence result for the corresponding local problem which can be considered as a limiting problem of (1) when α → 0, with k = F f . Berestycki and Lions [2,Theorem 1] proved that problem (4) has a ground state solution if nonlinearity k satisfies the assumptions (k 1 ) k ∈ C(R, R) is an odd function. Especially, for problem (1), Moroz and Van Schaftingen [18] supposed that nonlinearity f satisfies the assumptions (f 1 ) f ∈ C(R, R). (f 2 ) There exists C > 0 such that for every s ∈ R, |sf (s)| ≤ C(|s| f is an odd function and has constant sign on (0, +∞). Their result reads as follows Then problem (1) has a ground state solution. Furthermore, if f satisfies additional condition (f 5 ), then every ground state solution of problem (1) has constant sign and is radially symmetric with respect to some point in R N .
In this spirit, we are interested in the existence of ground state solutions for problem (1) with the upper critical nonlinear term in the present paper. For the critical case, we would like to mention [1,6,7,8] and the references therein. [1] obtained the existence, multiplicity and concentration behavior of the semi-classical solutions of problem (1) with singularly perturbation. [7,8] considered problem (1) in bounded domain. [6] considered problem (1) with periodic potential and nonlinearity. Based on the works above, we consider that f satisfies (f 6 ) f ∈ C(R, R) is an odd function and has constant sign on (0, +∞).
Here is our main result. Theorem 1.1. Assume that N ≥ 3, α ∈ (0, N ) and (f 6 ) − (f 9 ) hold. Then problem (1) has a ground state solution, which has constant sign and is radially symmetric with respect to some point in R N . Remark 1. In this paper, we study the autonomous Choquard equation (1) with the assumptions of Theorem 1.1 which has never been investigated. There are several difficulties in our paper. The main one is the reformative condition which means nonlinear term F is the upper critical growth. Contrasting with [18], this is what makes the present problem more complicated, and also means that their methods aren't suited to our problem. To get over these difficulties, we employ a similar argument in [2,15].
Remark 3. The main argument of the proof consists rather careful estimates the infimum of functional restrained on the Pohožaev manifold ( see Lemmas 2.2-2.5 in section 2 for detail ) which are much more precise than the ones seen so far. Besides, we are working on R N suggests that we may have to face a lack of compactness. We will overcome this difficulty by the compactness lemma of Strauss so that bounded Palais-Smale sequence has a converging subsequence. At last, the nonlinear term g in our paper need not be differentiable, then the constrained manifold need not be of class C 1 in our case, which implies that the minimizing sequence of the infimum isn't Palais-Smale sequence. Motivated by [2,15], we apply a new approach to seek Palais-Smale sequence ( see section 3 for detail ).
The present paper is organized as follows. In section 2 we give some preliminaries. Section 3 is devoted to the proof of Theorem 1.2.

2.
Preliminaries. From now on, we will use the following notations.
• H 1 (R N ) is the usual Sobolev space endowed with the usual norm • ·, · denotes action of dual.
• C, C i (i = 0, 1, 2, · · · ) denote various positive constants. From (f 6 ) − (f 8 ), for any δ > 0 there exists a constant C δ > 0 such that By the Hardy-Littlewood-Sobolev inequality [14] and Sobolev embedding, one has Then the energy functional I : This also implies that I is C 1 functional whose derivative is given by . Formally, the critical points of I are solutions for problem (1). We recall the Pohožaev manifold In order to complete the proof, we give the following lemmas.
there exists a unique t u such that u(xe −tu ) ∈ P and I (u(xe −tu )) = max t∈R I (u(xe −t )) ; (c) I is bounded from below on P by a positive constant.
The left hand side of (8) is continuous in We will take the function v in the family of On this family, we compute for every t, It is easy to see that there exists a constant it follows from the proof of (a) that there exists t u such that J(u tu ) = 0. Next we show the maximum point is unique.
For any t ∈ R, we set For the sake of convenience in writing, we define

GUI-DONG LI AND CHUN-LEI TANG
One can easily see that . One gets that there exists a unique t such that τ (t ) = 0, which means that τ (t) > 0 on (−∞, t ) and τ (t) < 0 on (t , +∞). Hence τ (t) is strictly increasing on (−∞, t ) and strictly decreasing on (t , +∞). Combining that τ (t) ≥ 0 as t → −∞, and τ (t) < 0 as t → +∞, thus there exists a unique t u such that τ (t u ) = 0, really, which is equivalent to that there exists a unique t u such that Φ (t u ) = 0. Moreover, (c) For u ∈ P, then J(u) = 0, it is quite clear that Combining (7) with (10), one obtains In fact, we only discuss one case that u < 1. Notice that u = 0 from the definition of P, combining with (11) , which means 1 ≤ C 1 u 2α N . Thus there exists ρ > 0 such that u ≥ ρ and ρ is independent of u. At the same time, it follows from (10) that It is quite clear that for any u ∈ P such that I(u) ≥ α 2(N +α) ρ 2 > 0.
Use S α to denote the best constant defined by From [6,7,8,14,19], we know that S α is achieved by where C is a fixed positive constant and ε > 0. Let ψ ∈ C ∞ 0 (R N , [0, 1]) be a cut-off function satisfying ψ = 1 for x ∈ Bρ and ψ = 0 for x ∈ R N \ B 2ρ = 0, whereρ is some positive constant. Define the test function by where u ε = ψU ε . According to [4,8,26], one obtains for ε small enough, and where l is some positive constant.
Then there exists a positive constant C 4 such that, for ε small enough, Proof. Let G + (s) = max{G(s), 0} and G − (s) = min{G(s), 0} for any s ∈ R. Indeed, from (7) and (13) we have for ε > 0 small enough, On the other hand, it follows from (g 1 ) − (g 3 ) that there exists C 2 such that which also implies that G − (s) ≥ −C 2 |s| N +α N for any s ∈ R. According to (3), (12) and (14), one obtains At the same time, from the Fubini theorem we have This completes the proof.
and for ε > 0 small enough there exists a positive constant C 8 such that Proof. Note that, for |x| < ε It follows from (g 3 ) that there exists A R > 0 such that for all s ∈ [A R , +∞), Combining with (16), when ε small enough one obtains for N ≥ 5, and for N = 3, By the arbitrariness of R, we get lim ε→0 |x|<ε 1 2

EXISTENCE OF GROUND STATE SOLUTIONS FOR CHOQUARD EQUATION 293
On the other hand, combining (3), (14) and (15) This completes the proof.
Proof. Note that, it follows from Lemmas 2.2 and 2.3 that for ε small enough, Then there exists an unique t vε such that v ε (xe −tv ε ) ∈ P from Lemma 2.1.
Proof. From Lemma 2.1, we can easily see that there exists a positive constant m such that m = inf u∈P I(u). Now, we define for t ∈ R, Then one can easily see that when t * = .

GUI-DONG LI AND CHUN-LEI TANG
At the same time, we recall that there exists t vε such that v ε (xe −tv ε ) ∈ P from (17). Combining with the Fubini theorem, we obtain According to Lemmas 2.2-2.4, we have m < S * for ε small enough. This completes the proof.
In order to prove the positive of solutions, the following propositions obtained by Moroz and Van Schaftingen [18] is of great significance, which relies on a nonlocal version of the Brézis-Kato estimate [3]. We give a sketch here for the reader's convenience and omit their proof.
3. Proof of Theorem 1.2. In order to complete the proof of Theorem 1.2, we are inspired by [4,15] to introduce the following equation where F q (t) = t 0 f q (s)ds and f q (s) = g(s) + q|s| q−2 s + pϕ(q)|s| p−2 s with p, q ∈ ( N +α N , 2 * α 2 ), ϕ(q) ≥ 0. Then the associated energy functional is and the corresponding Pohožaev manifold is We define it follows from (g 1 ) − (g 2 ) that there exists ϕ ∈ C(R + , R + ) such that f q ≥ 0 on (0, +∞) and ϕ(q) → 0 as q → 2 * α 2 . According to Theorem A, it is quite clear that there exists a positive radial ground state solution u q ∈ H 1 (R N ) of problem (18).
Proof. For any ε ∈ (0, 1 2 ), there exists u ∈ P such that I(u) < m + ε. Taking the function u in the family of functions u t ∈ H 1 (R N ) defined for t ∈ R and x ∈ R N by From lemma 2.1, there exists T ∈ R such that Since the continuity of R N (I α * F q (u t )) F q (u t )dx on (t, q) ∈ R × N +α N , 2 * α 2 , there exists θ > 0 such that for all 2 * α 2 − θ < q < 2 * α 2 and t ≤ T , which states that I q (u T ) ≤ − 1 2 for all 2 * α 2 − θ < q < 2 * α 2 . By Lemma 2.1, it is quite clear that there exists T 1 > |T | such that I q (u t ) > 0 for t < −T 1 . Therefore there exists t * q ∈ (−T 1 , T ) such that d dt I q (u t )| t=t * q = 0 and then u t * q ∈ P q . From Lemma 2.1, one has I(u t ) ≤ I(u). Thus for all Note that, for sequence {q n } ⊂ R, it follows from Theorem A that there exists a positive and radial symmetric sequence {u n } ⊂ H 1 (R N ) such that I qn (u n ) = 0 and I qn (u n ) = m qn , where q n ∈ ( N +α N , 2 * α 2 ) and q n → 2 * α 2 as n → +∞.
Proof. As a matter of fact, from (10) and Lemma 3.1, it is easy to see that which implies that {u n } is bounded in H 1 (R N ). Indeed, from lemma 2.1 there exists ρ > 0 such that u n ≥ ρ, we obtain Hence lim inf n→+∞ m qn > 0. This completes the proof.
To prove Theorem 1.2, we have to overcome the lack of compactness. Then the following two lemmas are central to our proof, which we sketch here for the readers' convenience. If one further assumes that P (s) Q(s) → 0 as |s| → 0 and u n (x) → 0 as |x| → +∞, uniformly with respect to n, then P (u n (x)) converges to v in L 1 (R N ) as n → +∞.
Lemma 3.4 (Strauss inequality, see [26]). If N ≥ 2, there exists C N > 0 such that Now we are ready to prove Theorem 1.2.
Proof of Theorem 1.2. Recall that {u n } ⊂ H 1 (R N ) in (19). Lemma 3.2 implies that {u n } is bounded in H 1 (R N ). Then there exists a subsequence, still denoted {u n }, such that • u n u ∈ H 1 (R N ); • u n → u ∈ L p (R N ) for any p ∈ (2, 2 * ); • u n (x) → u(x) a.e. in R N , where u ∈ H 1 (R N ) is a nonnegative and radially symmetric function obviously. For any φ ∈ C ∞ 0 (R N ), one has which means that u is a solution of problem (5). We claim that u = 0. By contradiction, suppose that u = 0. Set P (s) = |G(s)| Obviously, P (u n (x)) → 0 a.e. in H 1 (R N ) and R N Q(u)dx ≤ C. Moreover, Lemma 3.4 states that u n (x) → 0 as |x| → +∞ uniformly with respect to n. Then Lemma 3.3 implies that R N P (u n )dx → 0 as n → +∞. By the Hardy-Littlewood-Sobolev inequality and Sobolev embedding, we have as n → +∞, As the above process, set P 1 (s) = |g(s)s| 2N N +α , we also get that R N P 1 (u n )dx → 0 as n → +∞, and then R N (I α * |u n | qn )g(u n )u n dx = o(1).
Indeed, it is easy to see that as n → +∞, R N (I α * G(u n ))g(u n )u n dx = o(1).

By the Young inequality, one gets
Hence, for q n → 2 * α 2 as n → +∞, one sees
If lim n→+∞ R N |∇u n | 2 dx ≥ N −2 which contradicts Lemma 2.5. In either case, we arrive at a contradiction. So u = 0. Since I (u) = 0, by the weaker lower semi-continuity of the norm, we obtain m ≤ I(u) = 2 + α 2(N + α) R N |∇u| 2 dx + α 2(N + α) R N u 2 dx ≤ lim inf n→+∞ 2 + α 2(N + α) R N |∇u n | 2 dx + α 2(N + α) R N u 2 n dx = lim inf n→+∞ m qn ≤ m, which implies that I(u) = m. Moreover, u ∈ H 1 (R N ) is a nonnegative and radially symmetric function obviously. Then, to complete the proof of Theorem 1.2, we should seek that u is positive. Define H : R N → R and K : R N → R for x ∈ R N by H(x) = F (u(x)) u(x) and K(x) = f (u(x)). Observe that for every x ∈ R N , we have By Proposition 1, we obtain that u ∈ L p (R N ) for every p ∈ [2, N α 2N N −2 ). In view of (g 1 ) − (g 2 ), one gets F (u) ∈ L p for every p ∈ [ 2N N +α , N α 2N N +α ) (see [19,18]). It follows from the Hardy-Littlewood-Sobolev inequality that N +α , we have I α * F (u) ∈ L ∞ (R N ), and thus By the classical bootstrap method for subcritical local problems in bounded domains, we deduce that u ∈ L ∞ loc (R N ). Combining with the strong maximum principle [9,19,18], we know that u is positive. According to Proposition 2, we may conclude that u is a positive and radial ground state solution of problem (4). This completes the proof.