CONCENTRATION OF NODAL SOLUTIONS FOR SEMICLASSICAL QUADRATIC CHOQUARD EQUATIONS

. In this article concerns the semiclassical Choquard equation − ε 2 ∆ u + V ( x ) u = ε − 2 ( 1 |·| ∗ u 2 ) u for x ∈ R 3 and small ε . We establish the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function V , by means of the perturbation method and the method of invariant sets of descending ﬂow.


Introduction
In the past two decades, attention has devoted to the study of the existence, multiplicity, and properties of the solutions for the nonlinear Choquard equation where 0 < α < N , 2N −α N < p < 2N −α N −2 , and ε > 0 is a small positive parameter.When N = 3, α = 1 and ε = 1, as an important model, the problem was introduced by Pekar [30] to describe the quantum theory of a polaron at rest, and then used by Choquard [18] to study steady states of the one-component plasma approximation to the Hartree-Fock theory.Later, the same equation re-emerged as a model of self-gravitating matter [29], and in that context it is referred as to the Schrödinger-Newton system.
For the existence and qualitative properties of solutions for the nonlinear Choquard equation (1.1), we refer the reader to [2,3,5,9,10,14,17,26,27,31,32,34,35] and references therein.In particular, for p > 2, the existence of nodal solutions for the Choquard equation is an appealing aspect which is investigated in [5,8,11,13,15,16,23] by the variational method.In the physical case, for p = 2, the existence of nodal solutions for (1.1)only has few results.For p ≥ 2 and V is a radial symmetry function, Gui [13] show that, for any positive integer k, the equation(1.1)has a sign-changing solution u k which changes signs exactly k times.When V ≡ 1 and p = 2, Ghimenti [12] proved the existence of the least action nodal solutions.However, without symmetry or periodicity assumptions on the potential function V , there is no result of the existence of infinitely many signchanging solutions for the equation (1.1) with p = 2. Motivated by the works mentioned above, we consider the existence of infinitely sign-changing solutions for the following equation where the potential function V satisfies the assumptions: (A1) V ∈ C 1 (R 3 , R) and there exist constants b > a > 0 such that (A2) There exists a bounded domain M ⊂ R 3 with smooth boundary ∂M such that − → n (x), ∇V (x) > 0 , ∀ x ∈ ∂M , where − → n (x) is the outer normal of ∂M at x.
Under the assumption (A2), in view of the critical set without loss of generality we assume 0 ∈ A. For each set B ⊂ R N and any δ > 0, we set The main result of this paper reads as follows.
Theorem 1.1.Assume that (A1), (A2) hold.For each positive integer k, there exists 3) has at least k pairs of signchanging solutions ±v j,ε , j = 1, . . ., k.Moreover, for any δ > 0 there exist µ > 0, In this article, we can also obtain the existence and concentration phenomenon of the sign-changing solution of the equation(1.1).Here, we only consider the case with α = 1 and N = 3.
By making the change of variable εy = x, equation (1.3) is equivalent to and the corresponding functional is We will use the method of invariant sets of descending flow to prove the existence of sign-changing solutions for (1.3), but the setting of invariant sets of descending flow can not fit well for the Choquard equation.In [15], we used the perturbation method [24] to overcome this difficulty for Choquard equation (1.1) with 2 < p < 2N −α N −2 .However, the method described in [15] becomes invalid for the case p = 2.
To obtain compactness for the functional I ε , we use the penalization method in [4,36] ), we consider functionals: and (1.9) Note that and (1.11) We also note that the critical points of Γ ε and Γ ε,p are, respectively, solutions of ) t .We define (1.14) For each ϕ ∈ H 1 (R 3 ), since g λ (t)t + g λ (t) = b λ (t), we have (1.15) . By Hardy-Littlewood-Sobolev inequality we know that there exists C p > 0 such that ψ 1/2 (u) ≤ C p u p and C p independent of u.It is easy to know that when u ≤ ( 1 Cpλ ) ε,p (u).This article is organized as follows.In Section 2, we prove (P S) c condition for Γ ε,p and give some uniform estimates (independent of p) on the critical points of Γ ε,p .In Section 3, we prove the existence of sign-changing solutions for Γ ε .Section 4 is devoted to the proof of Theorem 1.1.
For a Banach space E, we denote its dual space by E .Throughout the paper, c, c 0 , c 1 , . . .denote different constants and c λ , C λ denote constants depending on λ.

(P S) c condition for Γ ε,p
In this section, we first collect elementary properties of the Choquard term, and then we prove that Γ ε,p satisfies the (P S) c condition.
Proof.By (1.14),(1.15)and Lemma 2.3, we have As a result, there exists a constant c L,p such that Lemma 2.5.For each L > 0, there exists ε L > 0 such that, for 0 < ε < ε L , if c < L, the following statements hold (1) The Γ ε,p satisfies the (P S) c condition. ( up to a subsequence. Proof. (1) The proof of this part is similar to that in [15].
(2) By (1.9) and (1.11), we have By (2.2) we know that there exists ηL > 0 independent of ε, p such that u n ≤ ηL and G( It is easy to show that u is a solution of the equation as n → ∞.Since u n ≤ ηL , we have u ≤ ηL .Hence, there exists a constant such that Using the dominated convergence theorem, we obtain as n → ∞.Then by (A1), u ≤ ηL and (2.5), we obtain (2.6) (2.7) By Fatou's Lemma, we have Using the interpolation inequality, for 2 < 6pn 5 < q < 2 * , we have where Combining with u n ≤ ηL , (2.7) and (2.8) we obtain where 12) The result of the lemma follows from (2.11) and (2.12).

Existence of sign-changing critical points for Γ ε
To obtain multiple sign-changing critical points of Γ λ ε,p , we introduce the abstract critical point theorem [22, Theorem 2.5], see also [6,Theorem 3.2].
Let X be a Hilbert space, f be an even C 2 -functional on X.Let P, Q be open convex sets of X, Q = −P.Set

Assume
(A3) there exists L > 0 such that f satisfies the (P S) c condition, for c < L; (A5) For every critical point x of f , D 2 f is a Fredholm operator.Also assume there exists an odd continuous map A : where γ is the genus of symmetric sets, γ(E) = inf n : there exists an odd map η : E → R n \{0} .
Assume that (A8) Γ j is nonempty.Define For u ∈ H 1 (R 3 ), we define v = Au by the unique solution to Note that A is odd, well defined, and continuous on This lemma can be proved as in [15,Lemma 3.2].Using Lemma 3.2, it is easy to prove assumption (A6).
Proof.For x ∈ M ε , we have χ ε (x) = 0 and V (εx) ≤ b.Then for u ∈ E j , we have As for [21, Lemma 5.6], we can complete the proof.
Proof.Note that every critical point of Γ ε,p is a weak solution of Assume u solves (3.3), by the sub-solution estimates, we have u ∈ L ∞ (R 3 ) and u(x) → 0 as x → ∞.For ψ, ϕ ∈ H 1 (R 3 ), we have ) is the Fredholm operator.On the other hand, the linear operator We define Moreover, there exists u j,ε,p ∈ K * cj (ε,p) such that m * (u j,ε,p ) ≥ j.
So there exists m > 0 such that u n H 1 (R 3 ) > m and 0 = u H 1 (R 3 ) ≥ m.Without loss of generality, we assume that u + = 0 and u − = 0. ( We define the normalized part as Then, up to a subsequence, there we have Corollary 4.3.There exist c, µ, p 1 , independent of n, such that, for 2 < p < p 1 , we have Proof.The Proof of Lemma 4.2 can be done by the same method as [15,Lemma 4.3].We only need to prove that there exists p 1 > 2 such that, for 2 < p < p 1 , By Moser's iteration, there exists a c > 0 such that u n,p L ∞ (R 3 ) ≤ c for 2 < p < p 0 .
By Lemma 4.2 and Hence, there exists p 1 > 2 such that for 2 < p < p 1 , we have co p (1) ≤ a 4 and Proof.If not, we assume that there exists i such that 1 ≤ i ≤ m and ε n > 0 such that lim n→∞ ε n = 0 and dist(y i , Ā) > 0. Let t k = ∇V (y i ) = 0, by (A2) we deduce that there exists δ 1 > 0 such that . By DΓ εn (u n ), ϕ = 0 for ϕ ∈ H 1 (R 3 ), we have where Next, we estimate all terms of (4.6).By (4.3), we have Hence the left-hand side of (4.6) is greater than or equal to Hence In particular there exists ε such that for 0 < ε ≤ ε we have Hence, where c 1 , c 2 > 0.
Proof.It suffices to prove that if X 0 is a subspace of H 1 (R 3 ) such that e −c2|x| ϕ(x)e −c2|y| ϕ(y) |x − y| dx dy where Similarly, we have Hence, we have By (4.13) and (4.15), we have, for ϕ ∈ X 0 , Hence, we obtain Now define the restriction operator P from L 2 (R 3 ) to L 2 (B R (0)) by P ϕ = ϕ| B R (0) .Since (4.16) holds, it is easy to see that P is injective.Let X0 = P X 0 , it suffice to prove X0 is finite-dimensional.It also follows from (4.16) that , then the set S is compact by (4.17).Hence, we obtain that X0 is finite-dimensional subspace.
The proof Theorem 1.1 follows from Propositions 4.6 and 4.8.