Role of sphingosine 1-phosphate (S1P) and effects of fingolimod, an S1P receptor 1 functional antagonist in lymphocyte circulation and autoimmune diseases

1.   Introduction and the main results

The fractional Laplacian operator $ (-\Delta)^s $ is defined by

where the symbol P. V. stands for the Cauchy principal value and $ C_{N, s} $ is a dimensional constant depending on $ N $ and $ s $, precisely given by

The nonlocal operators can be seen as the infinitesimal generators of Lévy stable diffusion processes ^[1]. Moreover, they allow us to develop a generalization of quantum mechanics and also to describe the motion of a chain or an array of particles that are connected by elastic springs as well as unusual diffusion processes in turbulent fluid motions and material transports in fractured media. The more physical background can be found in ^[9,10,16] and the references therein.

There are many papers considered the existence, multiplicity and qualitative properties of solutions for the fractional equations in the last decades, we refer to ^[2,7,8,11] for the subcritical case and to ^{[19,24,25,28]} for critical case, respectively. It is worth mentioning that some authors have been investigated the following Schrödinger equation

where $ V $ satisfies the following assumptions:

$ (V_1) $ $ V\in C(\mathbb{R}^N, \mathbb{R}) $ and $ V(x)\geq 0 $, $ \Omega: = int(V^{-1}(0)) $ is non-empty with smooth boundary.

$ (V_2) $ There exists $ M > 0 $ such that $ \big|\{x \in \mathbb{R}^N| V(x)\leq M \}\big| < \infty $, where $ |\cdot| $ denotes the Lebesgue measure.

Note that the function V satisfying $ (V_1) $ and $ (V_2) $ is called the deepening potential well, which was first proposed by Bartsch and Wang in ^[5]. When $ s = 1 $ and $ g(u) = |u|^{p-2}u $ with $ 2 < p < 2^* $, Bartsch and Wang ^[6] showed that, for $ \lambda $ large, (1.1) has a positive least energy solution, they also proved that a certain concentration behaviour of the solutions occur as $ \lambda\rightarrow\infty $. In^[13], Clapp and Ding actually generalized the results of ^[6] into the critical case. For more results to the Schrödinger equation with deepening potential well, we also cite ^{[3,4,21,25,26,27,31]} with no attempt to provide the full list of references.

Especially, if $ s\in(0, 1) $ and $ g(u) = (|x|^{-\mu} \ast F(u))f(u), $ then (1.1) goes back to the following fractional Choquard equation

There are many works involving the existence, multiplicity and qualitative properties for solutions of (1.2) in the recent periods, we can refer to ^{[12,14,18,24,30]} as well as to the references therein. Very recently, under the assumption of $ (V_1)-(V_2) $, Guo and Hu in ^[20] have proved the existence of the least energy solution to (1.2) with subcritical growth, which localizes near the bottom of potential well $ int(V^{-1}(0)) $ as $ \lambda $ large enough. It is a natural question that whether one can establish the similar results if nonlinearity is at critical growth, which inspired our present article. In this paper, we are concerned with the existence and asymptotic behavior of ground states for the following fractional Choquard equation with critical growth

where $ s\in (0, 1) $, $ N > 2s $, $ \mu\in (0, N) $, where $ 2^*_s = \frac{2N}{N-2s} $ is the fractional critical exponent, $ F(t) = \int_0^tf(s)ds $, $ f $ satisfies the following assumptions:

$ (f_1) $ $ f\in C^1(\mathbb{R}, \mathbb{R}) $, and there exist $ c_1 > 0 $ and $ \frac{2N-\mu}{N}\leq p_1\leq p_2 < \frac{2N-\mu}{N-2s} $ with $ p_1 > \frac{2N-\mu}{2N-4s} $ such that $ |f(t)|\leq c_1(|t|^{p_1-1}+|t|^{p_2-1}) $ for all $ t > 0 $.

$ (f_2) $ There exist $ q > 1 $ and $ c_2 > 0 $ such that $ f(t)\geq c_2|t|^{q-1} $ for all $ t > 0 $.

$ (f_3) $ $ \frac{f(t)}{t} $ is nondecreasing in $ (0, +\infty) $.

Remark 1.1. From $ (f_1)-(f_2) $, we have $ p_1\leq q\leq p_2. $ We point out that Ambrosetti-Rabinowitz condition is not necessary in present paper.

Remark 1.2. Taking $ f(t) = |t|^{p-2}t $, where $ p\in [\frac{2N-\mu}{N}, \; \frac{2N-\mu}{N-2s}) $ with $ p > \frac{2N-\mu}{2N-4s} $, then $ f $ satisfies $ (f_1)-(f_3) $. We also remark that besides the usual power function, there are many other functions that satisfy our assumptions. For example, we may choose suitable $ \mu $, $ s $, $ p $ and $ q $ such that $ 2\leq q \leq p < \frac{2N-\mu}{N-2s} $. By a direct calculation, the assumption $ (f_1)-(f_3) $ hold if we choose

To statement our main results of this paper, let us introduce the following fractional Choquard equation:

where $ s\in (0, 1) $, $ N > 2s $, $ \mu\in (0, N) $, which acts as a limit role for $ (Q_\lambda) $ as $ \lambda\rightarrow\infty. $ Our main results of this paper are stated as follows:

Theorem 1.1. Assume that $ (V_1)-(V_2) $ and $ (f_1)-(f_3) $ hold. Then, equation $ (Q_\lambda) $ has at least a positive ground state for $ \lambda $ large enough.

Theorem 1.2. Under the assumptions of Theorem 1.1, suppose that $ u_{\lambda_n} $ is one of the positive ground states of equation $ (Q_{\lambda_n}) $ with $ \lambda_n\rightarrow\infty $. Then, up to a subsequence, $ u_{\lambda_n}\rightarrow u $ in $ H^s(\mathbb{R}^N) $ as $ n\rightarrow\infty $. Moreover, $ u $ is a positive ground state of equation $ (Q_0) $.

In particular, by taking $ f(u) = |u|^{p-2}u $ in $ (Q_{\lambda}) $ and $ (Q_0) $, we obtain the following fractional Choquard equations:

and

where $ s\in (0, 1) $, $ N > 2s $, $ \mu\in (0, N) $.

As a direct result of Theorem 1.1 and Theorem 1.2, we have

Theorem 1.3. Assume that $ \mu\in (0, N) $ and $ (V_1)-(V_2) $ hold. Then, equation $ (P_\lambda) $ has at least a positive ground state for $ \lambda $ large enough if one of the following cases occurs:

$ (a) $ $ 2s < N < 4s $, $ p\in(\frac{2N-\mu}{2N-4s}, \frac{2N-\mu}{N-2s}) $.

$ (b) $ $ N = 4s $, $ p\in(\frac{2N-\mu}{N}, \frac{2N-\mu}{N-2s}) $.

$ (c) $ $ N > 4s $, $ p\in[\frac{2N-\mu}{N}, \frac{2N-\mu}{N-2s}) $.

Furthermore, suppose that $ u_{\lambda_n} $ is one of the positive ground states of equation $ (P_{\lambda_n}) $ with $ \lambda_n\rightarrow\infty $. Then, up to a subsequence, $ u_{\lambda_n}\rightarrow u $ in $ H^s(\mathbb{R}^N) $ as $ n\rightarrow\infty $. Moreover, $ u $ is a positive ground state of equation $ (P_0) $.

Remark 1.3. By Hardy-Littlewood-Sobolev inequality (see ^[22]), the energy functional corresponding to equation ($ P_\lambda $) belongs to $ C^1 $ if $ p\in [\frac{2N-\mu}{N}, \frac{2N-\mu}{N-2s}] $. However, we need to put further restriction on $ p $ to overcome the difficulties caused by the estimates of convolution term. It seems that the condition $ p > \frac{2N-\mu}{2N-4s} $ is essential for the proof of Lemma 8 below. Under the assumptions $ (V_1)-(V_2) $, whether or not the existence and asymptotic behavior of ground states of equation $ (P_\lambda) $ can be established is an interesting question for the case $ N = 4s $ with $ p = \frac{2N-\mu}{N} $ and the case $ 2s < N < 4s $ with $ p\in(\frac{2N-\mu}{N}, \frac{2N-\mu}{2N-4s}) $.

Compared with the nonlocal nonlinearity, the term $ (|x|^{-\mu} \ast F(u))f(u) $ depends not only the pointwise value of $ f(u) $, but also on $ |x|^{-\mu} \ast F(u) $, which leads to some estimates about nonlocal term are likely to be confronted with some difficulties. In order to overcome them, some new variational techniques will be employed in our paper. Another difficulty of the problem $ (Q_\lambda) $ stems from that we can not verify that the energy functional corresponding to equation ($ Q_\lambda $) satisfies the $ (PS)_c $ condition under the any level set due to the fact that $ H^s(\mathbb{R}^N)\hookrightarrow L^{2^*_s}(\mathbb{R}^N) $ is noncompact. On the contrary, we can only check that the functional satisfies the $ (PS)_c $ condition under a certain level set. Consequently, we have to make some more precise estimations involving critical term and nonlocal term.

The paper is organized as follows. In Section 2, we will introduce the variational frame and prove several Lemmas. In Section 3, we focus on the proofs of the main results.

Notation. Throughout this paper, $ \rightarrow $ and $ \rightharpoonup $ denote the strong convergence and the weak convergence, respectively. $ | \cdot|_r $ denotes the norm in $ L^r(\Omega) $ for $ 1\leq r\leq \infty $. $ B_\rho(x) $ denotes the ball of radius $ \rho $ centered at $ x $. $ C $ denote various positive constants whose value may change from line to line but are not essential to the analysis of the proof.

2.   Variational frame and some Lemmas

Before proving our main results, it is necessary to introduce some useful definitions and notations. Firstly, fractional Sobolev spaces are the convenient setting for our problem, so we will give some stretches of the fractional order Sobolev spaces. We recall that, for any $ s\in(0, 1) $, the fractional Sobolev space $ H^s(\mathbb{R}^N) = W^{s, 2}(\mathbb{R}^N) $ is defined as follows:

whose norm is defined as

where $ \mathcal{F} $ denotes the Fourier transform. We also define the homogeneous fractional Sobolev space $ \mathcal{D}^{s, 2}(\mathbb{R}^N) $ as the completion of $ \mathcal{C}_0^\infty(\mathbb{R}^N) $ with respect to the inner

and the norm

The embedding $ \mathcal{D}^{s, 2}(\mathbb{R}^N)\hookrightarrow L^{2^*_s}(\mathbb{R}^N) $ is continuous and for any $ s\in(0, 1) $, there exists a best constant $ S_s > 0 $ such that

The fractional laplacian, $ (-\Delta)^s u $, of a smooth function $ u: \mathbb{R}^N\rightarrow \mathbb{R} $, is defined by

Also, by the Plancherel formular in Fourier analysis, we have

As a consequence, the norms on $ H^s(\mathbb{R}^N) $ defined below

are equivalent. For more details on fractional Sobolev spaces, we refer the reader to ^[15] and the references therein. In this paper, the definition of fractional Sobolev space $ H^s(\mathbb{R}^N) $ is chosen by

equipped with the inner

whose associated norm we denote by $ \|\cdot\| $. Now, for fixed $ \lambda > 0 $, we define the following fractional Sobolev space

equipped with the inner product

whose associated norm we denote by $ \|\cdot\|_\lambda. $ Define

Obviously, $ E_0 $ is a closed subspace of $ H^s(\mathbb{R}^N) $, and hence is a Hilbert space.

Lemma 2.1. ^[25] Let $ 0 < s < 1 $, then there exists a constant $ C = C(s) > 0 $, such that

for any $ u\in H^s(\mathbb{R}^N) $. Moreover, the embedding $ H^s(\mathbb{R}^N)\hookrightarrow L^r(\mathbb{R}^N) $ is continuous for any $ r\in[2, 2^*_s] $ and is locally compact whenever $ r\in [1, 2^*_s) $.

Because we are concerned with the nonlocal problems, we would like to recall the well-known Hardy-Littlewood-Sobolev inequality.

Lemma 2.2. ^[22] Suppose $ \mu\in(0, N) $, and $ s, r > 1 $ with $ \frac{1}{s}+\frac{1}{r} = 1+\frac{\mu}{N} $. Let $ g\in L^s(\mathbb{R}^N) $, $ h\in L^r(\mathbb{R}^N) $, there exists a sharp constant $ C(s, \mu, r, N) $, independent of $ g $ and $ h $, such that

Since we are looking for ground states of $ (Q_\lambda) $ when $ \lambda $ is large enough, without loss of generality, we assume $ \lambda\geq 1 $ in the rest of the paper. We have the following embedding result.

Lemma 2.3. Assume that $ V(x) $ satisfies $ (V_2) $. Then the embedding $ E_\lambda\hookrightarrow H^s(\mathbb{R}^N) $ is continuous for any $ \lambda\geq 1 $. Moreover, there exists $ \tau_0 $ independent of $ \lambda $ such that

for any $ u\in E_\lambda $.

Proof. Let

For $ \lambda\geq 1 $, we have

By $ (V_2) $, the Hölder inequality and Lemma $ 2.1 $, one has

Consequently,

The proof is completed.

Since our main aim is to find the positive solutions, without loss of generality, we assume that $ f(t) = 0 $ for $ t\leq 0 $. The corresponding energy functionals associated with equations ($ Q_\lambda $) and ($ Q_0 $) are given by

and

respectively. Clearly, $ I_\lambda \in C^1(E_\lambda, \mathbb{R}) $ and $ I_0 \in C^1(E_0, \mathbb{R}) $. Denote

where

Remark 2.1. Obviously, $ u $ is a critical point of $ I_\lambda $ if and only if $ u $ is a solution of $ (Q_\lambda) $. Similarly, $ u $ is a critical point of $ I_0 $ if and only if $ u $ is a solution of $ (Q_0) $. Hence, in order to prove Theorem 1.1 and Theorem 1.2, it suffices to check that $ m_\lambda $ is achieved by a positive critical point of $ I_\lambda $ for $ \lambda $ large enough. Furthermore, for any sequence $ \lambda_n\rightarrow\infty $, if $ u_{\lambda_n} $ be one of the critical points of $ I_\lambda $, then there exists $ u\in H^s(\mathbb{R}^N) $ such that $ I_0^\prime(u) = 0 $ and $ I_0(u) = m_0 $. Moreover, up to a subsequence, $ u_{\lambda_n}\rightarrow u\; \; in\; \; H^s(\mathbb{R}^N) $.

Lemma 2.4. Let $ c > 0 $ be fixed. Assume that $ \{u^\lambda_n\} \subset E_\lambda $ be a $ (PS)_c $ sequence of $ I_\lambda $. Then

where $ \kappa_s = \min\{2^*_s, 4\} $. Moreover, there exist $ \delta > 0 $ independent of $ \lambda $ such that either $ u^\lambda_n\rightarrow0 $ in $ E_\lambda $ or $ \limsup_{n\rightarrow\infty}\|u^\lambda_n\|_\lambda > \delta $.

Proof. By $ (f_3) $, $ F(t)\leq 2f(t)t $ for any $ t\in\mathbb{R} $. Since $ I_\lambda^\prime(u^\lambda_n) = o_n(1) $ and $ I_\lambda(u^\lambda_n) = c+o_n(1) $,

Hence $ \{u^\lambda_n\} $ is bounded in $ E_\lambda $, and hence

This leads to

For any $ u\in E_\lambda $, by the Hardy-Littlewood-Sobolev inequality and Lemma $ 2.3 $, we have

Consequently, there exist $ \delta > 0 $ such that $ u\in E_\lambda $ with $ \|u\|_\lambda\leq \delta $, we have

If $ \limsup_{n\rightarrow\infty}\|u^\lambda_n\|_\lambda\leq \delta, $ without loss of generality, we may assume $ \|u^\lambda_n\|\leq \delta $ for all $ n $. By $ (2.6) $, one has

and hence $ \|u^\lambda_n\|\rightarrow 0 $ as $ n\rightarrow\infty. $

Lemma 2.5. Let $ C_0 > 0 $ be fixed, $ u^\lambda_n\rightharpoonup u_\lambda $ in $ E_\lambda $ with $ I(u^\lambda_n)\in[0, C_0] $. Then for any small $ \varepsilon > 0 $, there exists $ \Lambda_\varepsilon > 0 $ such that

for any $ \lambda > \Lambda_\varepsilon $ and $ 2\leq r < 2^*_s $.

Proof. Firstly, we claim that for any $ \varepsilon > 0 $, there exists $ \Lambda_\varepsilon > 0 $ such that

for any $ \lambda > \Lambda_\varepsilon $. We argue by contradiction that there exist $ \varepsilon_0 > 0 $, $ \lambda_k\rightarrow +\infty $ and $ n_k\rightarrow +\infty $ such that

Let $ D_R = \{x \in \mathbb{R}^N||x| > R \; \hbox{and}\; V(x)\leq M\}. $ In view of $ (V_2) $, $ \lim_{R\rightarrow\infty}\big|D_R\big| = 0. $ For $ k $ large enough, by (2.3) and the fact that $ \mathcal{D}^{s, 2}(\mathbb{R}^N)\hookrightarrow L^{2^*_s}(\mathbb{R}^N) $ is continuous, one has

It follows from (2.3) that

By (2.8)-(2.9), there exist $ K > 0 $ and $ R > 0 $ such that

Similarly, one can check that

Since $ u^\lambda_n\rightarrow u_\lambda\; $ in $ L^r_{loc}(\mathbb{R}^N) $ for $ 1\leq r < 2^*_s $, we may assume that

Combining (2.7) and (2.10)-(2.12), one has

a contradiction. For small $ \varepsilon > 0 $ and $ \lambda > \Lambda_\varepsilon $, by the interpolation inequality, we have

where $ 2\leq r < 2^*_s $.

Lemma 2.6. Let $ \lambda $ be fixed and $ \{u^\lambda_n\}\subset E_\lambda $ be $ (PS)_c $ of $ I_\lambda $. Then, there exists $ u_\lambda\in E_\lambda $ such that $ I_\lambda^\prime(u_\lambda) = 0 $ and $ I_\lambda(u_\lambda)\geq 0 $. Moreover, we have

and

where $ v^\lambda_n: = u^\lambda_n-u_\lambda $.

Proof. The proof is similar to ^[23]. For convenience sake, we give an outline here. For the sake of simplicity of symbols, we denote $ u^\lambda_n $ by $ u_n $. Lemma $ 2.4 $ implies that $ \{u_n\} $ is bounded in $ E_\lambda $. Up to a subsequence, we may assume that

It is easy to prove that $ I_\lambda^\prime(u_\lambda) = 0. $ Similar to (2.4), one has $ I_\lambda(u_\lambda)\geq 0 $. As the proof of the Lemma $ 2.4 $ in ^[23], we have the following nonlocal Brézis-Lieb result

It follows from Brézis-Lieb Lemma (see Lemma 1.32 in ^[29]) that

Combining (2.15) and (2.16), one has

Similarly, (2.14) is satisfied with some slight modifications.

Lemma 2.7. If $ c < \frac{s}{N}S_s^\frac{N}{2s} $, then there exists $ \Lambda_0 > 0 $ such that $ I_\lambda $ satisfies the $ (PS)_c $ condition for $ \lambda \geq\Lambda_0 $.

Proof. Consider any sequence $ \{u^\lambda_n\}\subset E_\lambda $ satisfying $ I_\lambda^\prime(u^\lambda_n)\rightarrow 0 $ with $ I_\lambda(u^\lambda_n)\rightarrow c < \frac{s}{N}S_s^\frac{N}{2s} $. By Lemma $ 2.4 $, $ \{u^\lambda_n\} $ is bounded in $ E_\lambda $. Let $ v^\lambda_n = u^\lambda_n-u_\lambda $. Then

By (2.16), (2.18) and Lemma $ 2.6 $, one has

Hence, up to a subsequence, we may assume

It suffices to check that there exists $ \varepsilon_0 > 0 $ such that $ \theta_\lambda = 0 $ for $ \lambda > \Lambda_{\varepsilon_0} $, where $ \Lambda_{\varepsilon} $ is given in Lemma $ 2.5 $. Otherwise, without loss of generality, there exists $ \lambda_k\geq\Lambda_{\frac{1}{k}}\geq 1 $ such that $ \theta_{\lambda_k} > 0 $ for any $ k\in \mathbb{Z} $. For large $ k $ and $ n $, by Lemma $ 2.5 $ and the Hardy-Littlewood-Sobolev inequality, one has

By Lemma $ 2.6 $, $ \{v^{\lambda_k}_n\} $ be $ (PS)_{c_k} $ for $ I_{\lambda_k} $, where $ c_k = c-I_{\lambda_k}(u_{\lambda_k}) $. Since $ \theta_{\lambda_k} > 0 $, by Lemma $ 2.4 $, we may assume that $ \theta_{\lambda_k}\geq\delta $ for all $ k $. By the definition of $ S_s $, there holds

Hence

and hence $ \theta_{\lambda_k}\geq S_s^\frac{N}{2s}(1-\frac{1}{\delta k})^\frac{N}{2s}. $ For large $ k $, by Lemma $ 2.6 $ and (2.19), one has

This leads to $ c\geq\frac{s}{N}S_s^\frac{N}{2s} $, which contradicts $ c < \frac{s}{N}S_s^\frac{N}{2s} $. This completes the proof.

Lemma 2.8. If $ p_1\in[\frac{2N-\mu}{N}, \frac{2N-\mu}{N-2s}) $ with $ p_1 > \frac{2N-\mu}{2N-4s} $, then there exists $ \alpha > 0 $ such that $ \alpha \leq m_\lambda\leq m_0 < \frac{s}{N}S^{\frac{N}{2s}}. $

Proof. Clearly, $ m_\lambda\leq m_0 $. Since the proof of $ m_\lambda \geq \alpha $ is standard, we only need to prove that $ m_0 < \frac{s}{N}S^{\frac{N}{2s}} $. Without loss of generality, we assume that $ 0\in\Omega $. Then there exist $ \delta > 0 $ and $ k\in \mathbb{Z} $ such that $ B_\delta\subset B_{2\delta}\subset\Omega\subset B_{k\delta} $. Let $ \eta\in C_0^\infty(\mathbb{R}^N) $ be such that $ 0\leq\eta\leq 1 $, $ \eta = 1 $ in $ B_\delta $, $ \eta = 0 $ in $ \mathbb{R}^N\setminus B_{2\delta} $. Denote

where $ u_0(x) = \alpha(\beta^2+S_s^{-\frac{1}{2s}}|x|^2)^{-\frac{N-2s}{2}} $ with $ \alpha, \beta > 0 $. Set

then $ u_\varepsilon(x)\in E_0 $. It follows from Proposition $ 21 $ and Proposition $ 22 $ in ^[25] that

Let

In view of (2.20), one has

Clearly, there exists $ t_\varepsilon > 0 $ such that $ t_\varepsilon u_\varepsilon \in \mathcal{N}_0 $ and $ I_0(t_\varepsilon u_\varepsilon) = \max_{t\geq0}I_0(t u_\varepsilon) $. As a consequence, $ m_0\leq I_0(t_\varepsilon u_\varepsilon) $ and

Next, we prove the following claim:

Claim 2.1.

.

In fact, by $ (f_2) $, for small $ \varepsilon > 0 $, we have

where $ c_1 $ and $ c_2 $ are given by $ (f_1) $. Since $ p_1 > \frac{2N-\mu}{2N-4s} $, $ N-1-\frac{2p_1N(N-2s)}{2N-\mu} < -1 $. Consequently,

By the Hardy-Littlewood-Sobolev inequality, we have

Similarly, one can check that

and

Since $ p_1\leq p_2 $, the claim follows from (2.24), (2.26)-(2.28).

For small $ \varepsilon > 0 $, by (2.21) and (2.23), there exist $ C_9 $, $ C_{10} > 0 $ such that

and

According to (2.22), we have

Thus, for small $ \varepsilon > 0 $ there exists $ t_0 > 0 $ such that $ t_\varepsilon\geq t_0 $. On the other hand, by $ (f_2) $, there holds

Hence

Since $ N > 2s $ and $ q\geq p_1 > \frac{2N-\mu}{2N-4s} $, then $ q > \frac{N+2s-\mu}{N-2s} $. Combining (2.21) and (2.29), one has

The proof is completed.

3.   The proof of the main results

3.1.   The proof of Theorem 1.1

Proof. Assume that $ \{u^\lambda_n\}\subset \mathcal{N}_\lambda $ be a minimizing sequence of $ m_\lambda $. By Ekeland's Variational principle (see^[17]), we may assume that $ \{u^\lambda_n\} $ be a $ (PS)_{m_\lambda} $ sequence for $ I_\lambda $, that is $ I_\lambda^\prime(u^\lambda_n)\rightarrow 0 $ and $ I_\lambda(u^\lambda_n)\rightarrow m_\lambda $. In view of Lemma $ 2.8 $, $ m_\lambda < \frac{s}{N}S_s^\frac{N}{2s} $. By lemma $ 2.7 $, there exist $ \Lambda_0 > 0 $, up to a subsequence, $ u^\lambda_n\rightarrow u_\lambda $ in $ E_\lambda $ for any $ \lambda > \Lambda_0. $ Since $ I_\lambda \in C^1(E_\lambda, \mathbb{R}) $, then $ I_\lambda(u_\lambda) = m_\lambda $ and $ I_\lambda^\prime(u_\lambda) = 0 $. Noting that $ f(t) = 0 $ for $ t\leq0 $ and $ (t-s)(t^-s^-)\geq|t^-s^-|^2 $ for all $ t\;, s\in\mathbb{R} $, one has

Thus $ u_\lambda\geq0 $. By Lemma $ 2.8 $, we have $ u_\lambda\neq0. $ In view of the Harnack inequality, $ u_\lambda > 0 $ and the proof is completed.

3.2.   The proof of Theorem 1.2

Proof. Suppose that $ \lambda_n\rightarrow\infty $ and $ u_{\lambda_n} $ be one of the ground states of equation $ (Q_{\lambda_n}) $. That is, $ I_{\lambda_n}(u_{\lambda_n}) = m_{\lambda_n} $ and $ I_{\lambda_n}^\prime(u_{\lambda_n}) = 0. $ We denote $ u_{\lambda_n} $ by $ u_n $ for notion simplicity. Without loss of generality, we assume that $ \lambda_n\geq 1 $ for all $ n $. As the proof of (2.4), one has

Hence $ \{u_n\} $ is bounded in $ H^s(\mathbb{R}^N) $. Up to a subsequence, we may assume that

We divide into four steps to prove Theorem $ 1.2 $ as follows.

Step 1: $ u(x) = 0 $ a.e in $ \mathbb{R}^N\setminus\Omega. $

If fact, by using the Fatou's Lemma, we get

which implies that $ u(x) = 0 $ a.e in $ \mathbb{R}^N\setminus \Omega $.

Step 2: $ u $ is a critical point of $ I_0 $.

Since $ I_{\lambda_n}^\prime(u_{\lambda_n}) = 0, $

It is clear that

By (3.1), we have

It is standard to prove that

and

Combining with the above results, we have $ I_0^\prime(u) = 0 $.

Step 3: $ u_n\rightarrow u $ in $ L^s(\mathbb{R}^N) $ for $ 2\leq s < 2^*_s. $

Similar to (2.8) and (2.9), one has

Hence, for any $ \varepsilon > 0 $ there exist $ R_1 = R_1(\varepsilon) > 0 $ such that

By the decay of the Lebesgue integral, there exists $ R_2 = R_2(\varepsilon) > 0 $ such that

By (3.1), one has

where $ R = \max\{R_1, R_2\} $. Consequently, $ u_n\rightarrow u $ in $ L^2(\mathbb{R}^N) $. By the interpolation inequality and the boundedness of $ \{u_n\} $ in $ H^s(\mathbb{R}^N) $, we have $ u_n\rightarrow u $ in $ L^r(\mathbb{R}^N) $ for $ 2\leq r < 2^*_s $.

Step 4: $ m_0 = I_0(u) $ and $ u_n\rightarrow u $ in $ H^s(\mathbb{R}^N) $.

By the Hardy-Littlewood-Sobolev inequality and the Lebesgue dominant convergence theorem, we get

and

It follows from the lower semicontinuity and the Fatou's Lemma that

As a consequence, $ I_{0}(u) = m_0 $ and $ [u_n]\rightarrow [u] $. By Step $ 3 $, $ \|u_n\|\rightarrow \|u\| $. This together with $ u_n\rightharpoonup u $ in $ H^s(\mathbb{R}^N) $, we have $ u_n\rightarrow u $ in $ H^s(\mathbb{R}^N) $. By Lemma 2.8, $ u\geq0 $ and $ u\neq0 $. According to the Harnack inequality, we have $ u > 0 $. The proof is completed.

3.3.   The proof of Theorem 1.3

Proof. Theorem $ 1.3 $ is directly concluded by Theorem $ 1.1 $ and Theorem $ 1.2 $.

From the proof of Theorem 1.2, we immediately get the following two Corollaries.

Corollary 3.1. $ m_\lambda\rightarrow m_0 $ as $ \lambda\rightarrow \infty. $

Corollary 3.2. Let $ \{u_{\lambda_n}\} $ be a solutions of equation $ (Q_{\lambda_n}) $ with $ \lambda_n\rightarrow\infty $ satisfying $ |I_{\lambda_n}(u_n) | \leq K $. Then up to a subsequence, $ u_n\rightarrow u $ in $ H^s(\mathbb{R}^N) $ as $ n\rightarrow\infty $. Moreover, $ u $ is a solution of equation $ (Q_0) $.

4.   Conclusions

In this paper, we are concerned with a fractional Choquard equation with critical growth. Under some assumptions of nonlinearity, we obtain the existence and asymptotic behavior of the positive ground states to this problem by applying some analytical techniques. Several recent results of the literatures are extended and improved.

Acknowledgments

This work is supported partially by NSFC (No. 11861078, 11971485, 11901345, 11661083), Hunan Provincial Innovation Foundation for Postgraduate and the Fundamental Research Funds for the Central Universities of Central South University.

Conflict of interest

The authors declare that they have no conflicts of interest.