MULTIPLICITY AND CONCENTRATION OF SOLUTIONS FOR CHOQUARD EQUATION VIA NEHARI METHOD AND PSEUDO-INDEX THEORY

. This paper concerns the following nonlinear Choquard equation: where ε > 0 , N > 2 , I θ is the Riesz potential with order θ ∈ (0 ,N ) , p ∈ (cid:2) 2 , N + θ N − 2 (cid:1) , min V > 0 and inf W > 0. Under proper assumptions, we explore the existence, concentration, convergence and decay estimate of semiclassical solutions for ( ∗ ). The multiplicity of solutions is established via pseudo-index theory. The existence of sign-changing solutions is achieved by minimizing the energy on Nehari nodal set.

1. Introduction and main results. We study the multiplicity of semiclassical solutions and the concentration phenomenon, convergence, decay estimate of groundstate solutions for a nonlinear Choquard equation. More precisely, we are interested in the following equation: − ε 2 ∆w + V (x)w = ε −θ W (x)(I θ * (W |w| p ))|w| p−2 w, w ∈ H 1 (R N ), (1.1) where ε > 0, N > 2, θ ∈ (0, N ), p ∈ 2, N +θ N −2 , V and W are bounded positive functions, and the Riesz potential I θ is defined as follows: . (1. 2) The Choquard equation once appeared in Fröhlich and Pekar's model of polaron [24] and was afterwards introduced by Ph. Choquard in 1976 in the modelling of a one-component plasma [15]. It can be regarded as the Schrödinger-Newton equation in models coupling the Schrödinger equation of quantum physics with nonrelativistic Newtonian gravity [4]. Moreover, it also associates with the Einstein-Klein-Gordon and Einstein-Dirac system [12]. Thereupon, Choquard equation has a rich Physical background.
Solitary wave solution with the type ψ(x, t) = e −it/ε w(x) for the focusing time-dependent Hartree equation with (t, x) ∈ R + × R N , iε ∂ψ ∂t = −ε 2 ∆ψ + (V (x) + 1)ψ − ε −θ W (x)(I θ * (W |ψ| p ))|ψ| p−2 ψ corresponds to the solution of Eq.(1.1). In this sense, Choquard equation is known as stationary Hartree equation. This is one of the motivations for the study of Eq.(1.1). As early as in 1977, E.H. Lieb [15] dealt with the following Choquard equation where a > 0, and he proved the existence and uniqueness (up to translations) of solutions by using symmetric decreasing rearrangement inequalities. Thereafter, P.L. Lions [17] also considered Eq.(1.3) and obtained the existence of infinitely many spherically symmetric solutions. From then on, Choquard equation has attracted more and more attention from researchers. For instance, Ackermann [1] studied the following equation where V ∈ L ∞ (R 3 ) was periodic separately in each coordinate direction with minimal period 1 and W ∈ L 1 (R 3 ) + L r (R 3 ) was nonnegative and even with r ≥ 1, he first proved a nonlinear superposition principle for zeros of equivariant vector fields that were asymptotically additive in a well-defined sense and then used this result to achieve the existence of multibump solutions for the equation. Wei and Winter [27] studied where ε > 0, V ∈ C 2 (R 3 ) and inf R 3 V (x) > 0, they proved that for any given positive integer K, if P 1 , P 2 , · · · , P K ∈ R 3 were given nondegenerate critical points of V , then for ε sufficiently small, there existed a positive solution for the equation and this solution had exactly K local maximum points Q ε i (i = 1, 2, · · · , K) with Q ε i → P i as ε → 0. Ma and Zhao [18] settled the longstanding open problem concerning the classification of all positive solutions to Eq.(1.3) with a = 1 and proved that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point by the method of moving planes. Moroz and Schaftingen [20] proved the existence of groundstate solutions for Eq.(1.1) with ε = V (x) = W (x) = 1, p ∈ N +θ N , N +θ N −2 and established the regularity, positivity, symmetry, monotonity and decay asymptotics of groundstate solutions. Moroz and Schaftingen [21] studied Eq.(1.1) with W (x) = 1, they assumed the external potential V ∈ C R N , [0, ∞) had a local minimum and obtained the existence of a family of positive solutions concentrating to the local minimum of V under optimal assumptions on the decay of V and admissible range of p ≥ 2 by using variational methods and nonlocal penalization technique. Alves, Cassani, Tarsi and Yang [2] considered the equation where ε > 0, θ ∈ (0, 2), V is continuous and F is the primitive of f , they assumed that f had a critical exponential growth in the sense of Trudinger-Moser and established the existence and concentration of positive groundstate solutions by variational methods. Bonheure, Cingolani and Schaftingen [6] studied the logarithmic Choquard equation where a > 0, they derived the sharp asymptotic decay and nondegeneracy of the unique positive radially symmetric groundstate solution.
Recently, the discussion about sign-changing solutions for Choquard equations has appeared increasingly. Clapp and Salazar [8] studied where Ω is an exterior domain in R N , N > 2, θ ∈ (0, N ), p ∈ 2, N +θ N −2 and V ∈ C(R N ), inf R N V > 0, lim |x|→∞ V (x) > 0, they established the existence of multiple sign-changing solutions with small energy in H 1 0 (Ω) under symmetry assumptions on Ω and V . Ghimenti and Schaftingen [10] studied Eq.(1.1) with ε = V (x) = W (x) = 1 and constructed minimal energy odd solutions for p ∈ N +θ N , N +θ N −2 and minimal energy nodal solutions for p ∈ 2, N +θ N −2 by introducing a new minimax principle for least energy nodal solutions and developing new concentration-compactness lemmas for sign-changing Palais-Smale sequences, it's interesting that the nonlinear Schrödinger equation as nonlocal counterpart of the Choquard equation does not have such solutions. Ruiz and Schaftingen [25] settled the open problem in [10] and proved that the least energy nodal solutions had an odd symmetry with respect to a hyperplane when θ was either close to 0 or close to N .
Ding and Wei [9] considered the following Schrödinger equation where ε > 0, p ∈ 2, 2N N −2 and V, W are continuous bounded positive functions, they studied the existence and concentration phenomena of semiclassical positive groundstate solutions. They also constructed the multiplicity of solutions including at least 1 pair of sign-changing solutions by pseudo-index theory and Nehari method.
Motivated by the works mentioned above, we intend to study the multiplicity and concentration of semiclassical solutions for the nonlinear Choquard equation (1.1). We shall use the ranges and interdependence of linear and nonlinear potentials to be the main assumptions. It is well known that the nonlocal convolution term with Riesz potential makes more difficulties in the process of discussion. We will prove that the convolution term and its derivative satisfy Brézis-Lieb type lemma. Finally we obtain the existence of multiple solutions, the least energy solutions and the least energy nodal solutions for Eq.(1.1). Moreover, we shall show the positive least energy solutions concentrate in a special set related to the minimum of linear potential and the maximum of nonlinear potential. Now we state our assumptions and main results.
(A0): V, W ∈ C 0,ι (R N , R) are bounded with some ι ∈ (0, 1), V attains a global minimum in R N with min R N V > 0, and W attains a global maximum in R N with inf R N W > 0.
To describe our results, denote Then for the maximal integer m ∈ N with (1.5) Eq.(1.1) possesses at least m pairs of solutions for small ε > 0. Furthermore, when m ≥ 2 and p ∈ 2, N +θ N −2 , among the solutions, at least one is positive, one is negative and two change sign. Theorem 1.2. Assume that (A0) holds and (1.6) Then for the maximal integer m ∈ N with all the conclusions of Theorem 1.1 keep true.
and for some r 0 ∈ (r, ∞), This paper is organized as follows: Section 2 offers some valuable information about the Riesz potential. Section 3 contains some preliminary results which are established by variational methods and play a key role in the arguments of main theorems. Section 4 and Section 5 contribute to the proofs of main results. In Section 4, we prove the multiplicity of solutions via Benci pseudo-index theory and show the existence of the least energy solutions and the least energy nodal solutions via Mountain Pass theorem and Location theorem, respectively. In Section 5, we discuss the convergence, concentration phenomenon and decay estimate of the positive least energy solutions.
2. Riesz potential. In this section, we shall recall and prove some useful results about the Riesz potential.
The Riesz potential with order θ ∈ (0, N ) of a function f ∈ L 1 loc (R N ) is defined as where A θ is the same as in (1. The Riesz potential I θ is well-defined as an operator in L q (R N ) if and only if q ∈ 1, N θ . Furthermore, if q ∈ 1, N θ and r := N q N −θq , then is a bounded linear operator, which can be disclosed by the following Hardy-Littlewood-Sobolev inequality.
N +θ (see [16]), then the optimal constant in above inequality is More generally, I θ could be interpreted as the inverse of the fractional Laplacian operator (−∆) θ 2 (see [14]). Applying Lemma 2.1 to the function f = |u| p ∈ L 2N N +θ (R N ) and by Hölder inequality, the following result is true.
It's wonderful that the Brézis-Lieb type lemma holds for Riesz potential. To achieve the proof of this lemma, we need two essential results.
Let Ω be a domain in R N , r ∈ [1, ∞) and {u n } be a bounded sequence in L r (Ω). If u n → u a.e. in Ω as n → ∞, then for any q ∈ [1, r], We would like to point out in advance that the constraint p ∈ 2, N +θ N −2 is only needed for the second part in the following Brézis-Lieb type lemma, while the first part permits p ∈ N +θ N , N +θ N −2 .
Proof. According to (2.18), we just need to show that for any v ∈ H 1 (R N ), 3. Preliminary results. In this section, we present some results which are necessary for the arguments of our main results. Let us first consider the following two Choquard equations for where a > 0, b > 0, and Associated with Eq.(3.1) and Eq.(3.2) respectively, we define the energy functionals for each v ∈ H 1 (R N ), the Nehari manifolds and the sets of least energy solutions The critical points of J ab and J ab ε correspond to the weak solutions of Eq.(3.1) and Eq.(3.2), respectively.
The proof of Lemma 3.2 can be easily finished by applying Lemma 2.2.
Proof. The proof is similar to that of Theorem 1 in Moroz and Schaftingen [20]. For completeness, we give the details. Firstly, we show that s ab is attained in as n → ∞, which contradicts with (3.6) by Lemma 2.2. Thus there exist δ > 0 and y n ∈ R N such that (3.7) Set u n := v n (· + y n ), then {u n } is also bounded in H 1 (R N ) and S ab (u n ) = S ab (v n ). Meanwhile, (3.6) deduces that Without loss of generality, we may assume that u n u in H 1 (R N ) as n → ∞. Then u n → u in L 2 loc (R N ) as n → ∞, which, together with (3.7), implies u = 0. Noting that we obtain If R N b 2 (I θ * |u| p )|u| p dx < 1, then it follows from (3.8), (3.9), (3.10), p ≥ 2 and Lemma 2.5-(i) that which is a contradiction. Thus R N b 2 (I θ * |u| p )|u| p dx = 1 and by (3.9), s ab = S ab (u).
Next we will show ϑ ab is attained in H 1 (R N ). According to Lemma 3.3 and Lemma 3.4, we get that

MIN LIU AND ZHONGWEI TANG
Noting that we obtain That is v ∈ R ab .
In view of Theorem 3 in Moroz and Schaftingen [20], we have the following lemma.
, u is either positive or negative, and u is radially symmetric up to translations. ( Lemma 3.8.
. One can check that v ∈ N ab if and only if u ∈ N ∞ , and for any v ∈ N ab , 3.2. The equation (3.2). In this subsection, we shall establish some results for Eq.(3.2). Lemma 3.9. There exist ρ > 0 and σ > 0 both independent of ε, a, b and just dependent on N, θ, p, τ, k such that (1)) < 0 . The proof of Lemma 3.10 is similar to that of Lemma 3.3 and so is omitted. (3.11) By slight amendment in the proof of Lemma 2.5, we get that and for any ϕ ∈ H 1 (R N ), (3.13) As the proof of Lemma 2.6, we get that for any ϕ ∈ H 1 (R N ), which ensures that (J ∞ ε ) (v) = 0. In virtue of (3.11), (3.12) and (3.13), we obtain that (3.17) Taking into account (3.15), we have which, together with (3.17), implies that If R N (I θ * |z n | p )|z n | p dx → 0 as n → ∞, then z n 1 → 0 as n → ∞. Thus v n → v in H 1 (R N ) as n → ∞ and c = J ∞ ε (v) ≥ ϑ ∞ ε . If R N (I θ * |z n | p )|z n | p dx ≥ δ > 0, then t n → 1 as n → ∞ by (3.18) . In virtue of Lemma 3.3 and Lemma 3.10, ϑ ∞ ε and ϑ ∞ are Mountain Pass levels of J ∞ ε and J ∞ , respectively. Therefore, ϑ ∞ ε ≥ ϑ ∞ . Remark 1. Similarly, if J ab ε has a (P S) c sequence, then either c = 0 or c ≥ ϑ ab ε .
then there exists ε ab > 0 such that for all ε ≤ ε ab , ϑ ab ε is attained at some v ab ε > 0. Proof. Noting Lemma 3.8 and (3.28), we have ϑ αβ < ϑ ∞ , where α = V a (0) and β = W b (0). By Lemma 3.13 and Lemma 3.11, ∃ε ab > 0 such that ϑ ab ε < ϑ ∞ ≤ ϑ ∞ ε , ∀ε ≤ ε ab . By Lemma 3.12, J ab ε satisfies (P S) ϑ ab ε condition for all ε ≤ ε ab , which, together with Lemma 3.9 and Lemma 3.10, implies that ϑ ab ε is attained at v ab ε ∈ H 1 (R N ). Since J ab ε (v) = J ab ε (|v|) for any v ∈ H 1 (R N ), we may assume v ab ε ≥ 0. By bootstrap method and elliptic regularity theory, v ab ε ∈ C 2 (R N ). By strong maximum principle, v ab ε > 0. 3.3. Preparation for sign-changing solutions. In order to prove the existence of sign-changing solutions, we define the Nehari nodal set , v ± = 0 and the least energy nodal value Indeed, N ab ε ⊃ M ab ε = ∅ and ζ ab ε ≥ ϑ ab ε > 0. Lemma 3.15. For any v ∈ H 1 (R N ) with v ± = 0, there exists a unique pair of (s v , t v ) ∈ R + × R + such that Proof. For any v ∈ H 1 (R N ) with v ± = 0, consider the mapping then there exist R > r > 0 such that for any s, t ∈ [r, R], Let B 2 denote the closed unit disc in the plane R 2 and deg be the classical topological degree of Brouwer. As similar as the proof of Proposition 3.2 in Ghimenti and Schaftingen [10], we have the following minimax principle. Lemma 3.16. If p ∈ 2, N +θ N −2 , then for any δ > 0, Moreover, ∀γ ∈ Γ, M ab ε ∩ γ(B 2 ) = ∅. Remark 2. The continuity of ξ on the subset of constant-sign functions in H 1 (R N ) is ensured by the Hölder inequality, the Hardy-Littlewood-Sobolev inequality (Lemma 2.1), the Minkowski inequality, the classical Sobolev inequality and the assumption p > 2.
4. Proofs of the multiplicity of solutions. In this section, we will prove the existence of multiple solutions, the least energy solutions and the least energy nodal solutions for Eq.(1.1).
Step 1. We shall construct a m-dimensional subspace E rm of H 1 (R N ) such that where r m and ε m are existing constants depending on m.
Taking into account that Krasnoselski genus satisfies dimension property(see Benci [5]), we get Noting (4.6) and Lemma 3.11, we get that for any r ≥ r m , ε ≤ ε m , Now we prove c j (1 ≤ j ≤ m) are critical values of J ε by applying the Theorem 1.4 in Benci [5]. Set Since J ε is an even functional, then (4.10) Assume by contradiction that (4.10) is false, then ∃v n / Letting n → ∞, we have That is {v n } is a (P S) c sequence of J ε . Without loss of generality, we may assume and (4.10) imply that Set η :=η(1, ·), then η is an odd homeomorphism of H 1 (R N ) and For any A ∈ Σ and A ⊂ (J ε ) c0 , it follows from Lemma 3.9-(i) that A ∩ ∂B ρ = ∅. Uniting (4.8), (4.9), (4.11), (4.12) and (4.13), we obtain that c j (1 ≤ j ≤ m) are critical values of J ε , and gen(K c ) ≥ r + 1 if c := c k = c k+1 = · · · = c k+r with k ≥ 1 and k + r ≤ m. Since J ε is even, we conclude that J ε has at least m pairs of critical points which are also solutions of Eq.(4.1).
Step 3. We will prove Eq.(4.1) has at least one positive and one negative least energy solutions.
Step 4. We will prove Eq.(4.1) has at least one pair of sign-changing solutions when m ≥ 2 and 2 < p < N +θ N −2 .
This completes the proof.
Proof of Theorem 1.2. We can assume without loss of generality that x w = 0. Then V (0) = τ w , W (0) = k. By (1.6) and (1.7), we get m ≥ 1. Taking a = τ w , b = k in Eq.(3.1), there exists v ∈ R τwk . The following arguments are similar to the proof of Theorem 1.1, so the details are omitted.

Proofs of convergence, concentration and decay estimate of solutions.
In this section, we shall prove the convergence, concentration and decay estimate of the positive least energy solutions for Eq.(1.1). Namely we give the proof of Theorem 1.3.
Step 1. We shall prove the convergence of v ε as ε → 0 up to a sequence after translations.
Let ε j → 0(j → ∞), v j := v εj ∈ R εj with v j > 0. Since it follows from Lemma 3.13 that {v j } is bounded in H 1 (R N ). By v j ∈ N εj and Lemma 2.2, we have v j which contradicts with (5.1). Therefore, there exist δ > 0 and y j ∈ R N such that (5.5) In view of the boundness of {u j }, we can assume without loss of generality that 7) which, together with (5.2), implies u = 0.
Since V and W are bounded, going if necessary to a subsequence, we assume V εj (y j ) → V 0 and W εj (y j ) → W 0 as j → ∞. ∇V (ε j y j + tε j x)ε j xdt ≤ ε j M r, ∀x ∈ B r (0).
That is u j → u in H 1 (R N ) as j → ∞.
If the thesis is false, jointly with sub-solution estimate, then ∃δ > 0, x n ∈ R N with |x n | → ∞, j n ∈ N and C > 0 independent of n such that Noting u jn → u in L 2 (R N ) as n → ∞ and by Minkowski inequality, one has That is a contradiction.
Step 3. We claim {ξ j y j } j is bounded in R N .
Therefore, going if necessary to a subsequence, we may assume ε j y j → x 0 as j → ∞. Step 4. We claim {εy ε } ε is bounded, where y ε ∈ R N is a maximum point of v ε .
Assume by contradiction that there is ε j → 0 with |ε j y j | → ∞, where y j := y εj is a maximum point of v j := v εj . Repeating Step 1, 2, 3, one can get that ∃y j ∈ R N such that u j = v j (· + y j ) → u = 0 in H 1 (R N ) as j → ∞, (5.26) u j (x) → 0 as |x| → ∞ uniformly in j ∈ N, Thus |ε j y j −ε j y j | ≥ |ε j y j |−|ε j y j | → ∞ as j → ∞, which implies that |y j −y j | → ∞ as j → ∞. Then max R N v j = v j (y j ) = u j (y j − y j ) → 0 as j → ∞. Hence max R N u j → 0 as j → ∞. Noting u j > 0, one has u j (x) → 0 as j → ∞ uniformly in x ∈ R N , which contradicts with (5.26).

By
Step 4, there exists ε j → 0 with ε j y j → y 0 as j → ∞, (5.27) where y j := y εj is a maximum point of v j := v εj . It is sufficient to check that where R :=R + sup ε |x ε |. The proof is completed.