EXISTENCE OF POSITIVE GROUND STATE SOLUTIONS FOR CHOQUARD EQUATION WITH VARIABLE EXPONENT GROWTH

. In this paper, we investigate the following Choquard equation − ∆ u = ( I α ∗ | u | f x | u | f 2 u in R N , where N ≥ 3, α ∈ (0 ,N ) and I α is the Riesz potential. If f ( x ) = (cid:40) p, x ∈ Ω , ( N + α ) / ( N , x N , where 1 < p < N + α N − 2 and Ω ⊂ R N is a bounded set with nonempty, we obtain the existence of positive ground state solutions by using the Nehari manifold.


Introduction. The stationary Choquard equation
−∆u + u = (I α * |u| p )|u| p−2 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 2 ) 2 α π N/2 Γ( α 2 ) and Γ is the Gamma function (see [37, p.19]), arises in many interesting physical situations in quantum theory and plays an important role in the theory of Bose-Einstein condensation where it accounts for the finite-range many-body interactions. Especially, N = 3 and α = p = 2 in equation (1), that is, It was investigated by Pekar in [34] to study the quantum theory of a polaron at rest. In [19], Choquard applied it as approximation to Hartree-Fock theory of onecomponent plasma. This equation was also proposed by Penrose in [28] as a model of self gravitating matter and was known in that context as the Schrödinger-Newton equation.
Mathematically, the existence of solutions for problem (2) has been studied with variational techniques by Lieb [19], Lions [21] and Menzala [27]. Ma and Zhao [25] investigated problem (2) and obtained the uniqueness of the positive solutions by the radial symmetry and monotone decreasing property of all positive solutions. Later, Wang and Yi [41] based on the classification results of Ma and Zhao, showed the uniqueness of the positive solutions of problem (2) for N = 3, 4, 5.
For problem (1), there exist lots of papers in which the authors considered the existence and qualitative properties of solutions. To illustrate, Moroz and Van Schaftingen [31] obtained the existence of a positive ground state solution for problem (1) with p ∈ ( N +α N , N +α N −2 ), then Ruiz and Van Schaftingen [38] proved that least energy nodal solution of problem (1) is odd with respect to a hyperplane of R N where α is either close to 0 or close to N and p ∈ [2, N +α N −2 ). The variants of problem (1) that include the introduction of an external potential and the treatment of more general nonhomogeneous nonlinearities instead of |u| p were also considered. Such as Alves, Nobrega and Yang [5] obtained Multi-bump solutions for problem (1) with deepening potential well (also see [24]), Van Schaftingen and Xia [40] showed the existence of a groundstate, of an infinite sequence of solutions of unbounded energy and the existence of least energy nodal solution with the coercive potential, Zhang et al. [43] assumed that a more weak condition than coercivity on potential, precisely, there exists a constant r > 0 such that, for any M > 0, meas {x ∈ R N : |x − y| ≤ r, V (x) ≤ M } → 0 as |y| → +∞. Moroz and Van Schaftingen [32] considered problem (3) with p = N +α N and an external potential, [30] studied problem (1) when the homogeneous nonlinearities |u| p is replaced by F (u), [13,44] studied problem (1) with an external potential and a more general nonlinearity G(x, u), [14,15] considered problem (1) in bounded domains, and so on. For a complete and updated discussion upon the current literature of such problems, we refer the interested reader to the guide [29]. We also mention [3,4,33], where the semiclassical case is considered.
In recent years, there has been an increasing interest in the study about problem of elliptic equation with variable exponent growth conditions, who are motivated by their applications to the mathematical models. For example, non-Newtonian fluids [8] (in particular, electro-rheological fluids [1]), nonlinear Darcy's law in porous medium [7], image processing [12]. For the precise applied background we refer to [9,34,36,45]. Moreover, the qualitative properties of solutions for these problems were considered by many authors. Further results for related problems can be founded in [2,6,11,17,18,23,26,35] and references therein. However, it is a new topic that the study on the existence of solutions for equations (3) up until now. Thus in this paper, we are interested in the existence of solutions for the following Choquard equation with variable exponent growth where f is a given function. Inspired by above papers, we assume that f satisfies the following conditions (f 1 ) there exists a bounded set Ω ⊂ R N with nonempty and 1 < p < N +α The exponent N +α N −2 (or N +α N ) is called the upper (or lower) critical exponent with respect to the Hardy-Littlewood-Sobolev inequality (see [20,21]).
Here we give our main result. Remark 1. In the present paper, we consider problem (3) in the whole space, generally speaking, whose difficulties lies in two main aspects. Firstly, the compactness fails in the whole space, and secondly D 1,2 (R N ) is embedded in L q (R N ) if and only if q = 2N N −2 , moreover, the embedding is not compact. Due to the lack of compactness, bounded (P S) m sequence could not converge, and the methods in [14,15] do not apply to Choquard equation (3) in the whole space. Under the assumption (f 1 ), the equation (3) exponent varies with the change of x, so it is not autonomous, and then it is meaningless that working in spaces of radial functions to recover the compactness by translations.
To overcome those difficulties, we will use some analysis methods and the Nehari manifold. Finally, we obtain the existence of positive ground state solutions of equation (3) through the critical point theory.

Remark 2. It is known that the functional associated with problem (3) is well defined if and only if
in order to control the nonlocal term in terms of the L 2 (R N ) norm (see [31]). However for our problem, the restriction N +α N < p isn't necessary and we only need 1 < p ≤ N +α N −2 from clear proof in Sectionen 2 to sure that the functional is well defined.
The present paper is organized as follows. In the next section we present some preliminary results. In Section 3, we give the proof of Theorem 1.1.

2.
Preliminaries. From now on, we will use the following notations.
• D 1,2 (R N ) is the completion of C ∞ 0 (R N ) with respect to the norm • L p (R N ) is the usual Lebesgue space endowed with the norm • meas Ω denotes the Lebesgue measure of the set Ω.

GUI-DONG LI AND CHUN-LEI TANG
there exists a sharp constant C(N, λ, q), independent of g and h, such that and . Then from (4), (5) and (6) we have where At the same time, combining (4) with the Hölder inequality, we also get Similarly, we have Then the energy functional I : D 1,2 (R N ) → R associated with problem (3) is well defined. The following Lemma 2.1 implies that I is C 1 functional whose derivative is given by We recall the Nehari manifold In order to complete the proof, inspired by [10,22] we introduce the following equation Then the associated energy functional is Now, giving the following lemmas to support the proof of Theorem 1.1.
Proof. Indeed, define In fact, we only need to prove . Given x ∈ R N and 0 < |t| < 1, from the mean value theorem, there exists θ 1 ∈ (0, 1) such that Hence, we have

GUI-DONG LI AND CHUN-LEI TANG
It follows from (4) that (13) is integrable in L 1 (R N ). From the Lebesgue dominated convergence theorem and the Fubini lemma that where Ω c = R N \ Ω. By almost the same process above, we have Assume that u n → u in D 1,2 (R N ), then u n → u in L 2 * (R N ) and L 2N p/(N +α) (Ω). From (7), (8) and (9), one notices that This completes the proof.
Then we have Combining with f (x) > 1, one can easily see that Ψ (t) > 0 as t small enough, and Ψ (t) < 0 as t large enough, which implies that there exists t > 0 such that I (tu), tu = 0, namely tu ∈ N . Moreover, the uniqueness of t is obvious.
(b) For u ∈ N , according to (4) and the Sobolev embedding, one obtains In fact, we only discuss the case that u < 1. Recall that u = 0 by the definition of the Nehari manifold. Since 2 < 2p < 2 * (N +α) N , there exists δ > 0 such that u ≥ δ and δ is independent of u.
(c) For any u ∈ N , one has It is quite clear that for any u ∈ N such that I(u) > 0 from (f 1 ).
(d) Considering the derivative of J(u) = I (u), u at u and applied in u, we obtain This completes the proof. Use S α to denote the best constant defined by From [14,15,31,39], we know that S α is achieved by where C N.α > 0 is a fixed constant. Proof. Recall some known facts, Fix v ∈ D 1,2 (R N ) and for s ∈ R small enough, say s ∈ (−ε, ε), the function φ + sv is not identically zero. Therefore there exists t : (−ε, ε) → (0, +∞) such that Notice that the application s → t(s)(φ + sv) defines a curve that passes through φ when s = 0. The function t is differentiable on (−ε, ε), and we have We define Since t(s)(φ + sv) for every s ∈ (−ε, ε), the point s = 0 is a local minimum for γ. The function φ is differentiable and So that Thus shown that for every v ∈ D 1,2 (R N ), Set φ = λω with λ ∈ R to be determined. Then ω satisfies , we see that ω is positive and satisfies namely ω = µφ is a positive solution of problem (11) where µ = S Proof. Firstly, we define for t ∈ (0, +∞), Then from lemmas 2.2 and 2.3, we can see that sup t∈(0,+∞) Ψ ∞ (t) is obtained by t = 1. This implies This completes the proof.
For the sake of convenience in writing, we define where x ∈ R N and x n := (0, 0, 0, · · · , 0, n) ∈ R N . Then ω n 2 = ω 2 and there exists ω ∈ D 1,2 (R N ) such that Since for any x ∈ R N , ω n (x) → 0 and ω = 0. Now for estimating the energy c, we give the following lemmas.
Proof. It is quite clear that there exists t n ≥ 0 such that t n ω n ∈ N from Lemma 2.2. Indeed, I (t n ω n ), t n ω n = 0 implies It follows from (17) and the fact that ω n is also a solution of problem (11) that which implies t n ≥ 1. Next, we seek lim n→∞ t n = 1.
Since Ω is bounded, there exists R such that Ω ⊂ B R . One can easily see that According to (7) and (18), we have At the same time, combining with (8) we also get According to (19) and (20), for n large enough one obtains

Thus one has
which implies that lim n→∞ t n = 1 from lemma 2.3. This completes the proof.
Lemma 2.6. Suppose that N ≥ 3, α ∈ (0, N ) and (f 1 ) holds. Then lim n→∞ Ω Ω I α (x − y)|ω n (y)| p |ω n (x)| p dydx Proof. Since the interior of Ω is nonempty, there exist z 0 ∈ R N and r > 0 such that Thus for n large enough one has Br(z0) Br(z0) At the same time, there exists R > 0 such that we also can easily see that Combining (21) and (22), one obtains lim n→∞ Ω Ω I α (x − y)|ω n (y)| p |ω n (x)| p dydx where C is a positive constant and independent of ω n . This completes the proof.
Lemma 2.7. Suppose that N ≥ 3, α ∈ (0, N ) and (f 1 ) holds. Then Proof. For n large enough, it is quite clear that At the same time that, we also can easily see that Combining (23) with (24), we obtain where C is a positive constant and independent of ω n . This completes the proof.
Proof. Recall that t n ω n ∈ N and there exists T * > 1 such that t n < T * for any n from Lemma 2.5. Thus combining with Lemmas 2.5-2.7, for n large enough one obtains c ≤I(t n ω n ) Therefore c < I ∞ (ω) = c ∞ . This completes the proof.
3. Proof of Theorem 1.1. To prove Theorem 1.1, we have to overcome the lack of compactness. Then the following lemma is central to our proof, which we sketch here for the readers' convenience. I satisfies the Palais-Smale condition at level m if {u n } has a converging subsequence. Note that, {u n } is bounded obviously. Thus there exists u ∈ D 1,2 (R N ), up to a subsequences, such that For any v ∈ D 1,2 (R N ), it is easy to see that Hence we obtain Define v n = u n − u. Thus one gets v n 2 + u 2 = u n 2 + o(1).

It follows from [31, Theorem 2.4] that
Combining with I(u n ) → m, we obtain At the same time, from I (u n ), u n = 0 and I (u), u = 0, one gets In fact, from 2N p N +α ∈ [1, 2 * ), one can easily see that for bounded Ω, Hence, according to proof of Lemma 2.5, (25) and (26) it is quite clear that From the definition of S α , we have Combining with (27) and (28), we see 2048 GUI-DONG LI AND CHUN-LEI TANG This deduces that either case 1. lim n→∞ R N |∇v n | 2 dx = 0, or case 2. lim n→∞ R N |∇v n | 2 dx ≥ S N +α α+2 α . If lim n→∞ R N |∇v n | 2 dx = 0, which implies v n → 0 in D 1,2 (R N ). This completes the proof. Therefore we only seek that the second case is never to happen.
If lim n→∞ R N |∇v n | 2 dx ≥ S which is a contradiction. Thus lim n→∞ R N |∇v n | 2 dx = 0, which deduces that u n → u ∈ D 1,2 (R N ). This completes the proof.
Now we ready to prove Theorem 1.1.
Proof of Theorem 1.1. By the Ekeland variational principle (see [42]), it is quite clear that there exists a Palais-Smale sequence {u n } ⊆ N for I at level c and u n (x) ≥ 0 a.e. in R N . Since c ∈ (0, c ∞ ) from lemmas 2.2 and 2.8, Lemma 3.1 implies that there exists u ∈ D 1,2 (R N ) such that, up to a subsequence, u n → u in D 1,2 (R N ). Thus that I (u) = 0 and I(u) = c. It follows from the definition of c that u is a ground state solution of problem (3). By the strong maximum principle [16,Theorem 8.19], we obtain that u is positive. This completes the proof.