Ground state sign-changing solutions for critical Choquard equations with steep well potential

. In this paper, we study sign-changing solution of the Choquard type equation where N ≥ 3, α ∈ (( N − 4 ) + , N ) , I α is a Riesz potential, p ∈ 2 , N N is the upper critical exponent in terms of the Hardy–Littlewood–Sobolev inequality, µ > 0, λ > 0, V ∈ C ( R N , R ) is nonnegative and has a potential well. By combining the variational methods and sign-changing Nehari manifold, we prove the existence and some properties of ground state sign-changing solution for λ , µ large enough. Further, we verify the asymptotic behaviour of ground state sign-changing solutions as λ → + ∞ and µ → + ∞ , respectively.


Introduction and main results
The Choquard equation has a physical prototype, namely the Hartree type evolution equation where R + = [0, +∞), I 2 (x) = 1 4π|x| , ∀ x ∈ R 3 \{0}, and * is convolution in R 3 . Eq. (1.1) was firstly proposed by Pekar to describe a resting polaron in [24]. Two decades later, Choquard [16] introduced Eq. (1.1) as a certain approximation to Hartree-Fock theory of one component plasma, and used it to characterize an electron trapped in its own hole. Afterwards, viewing the quantum state reduction as a gravitational phenomenon in quantum gravity, Penrose et al. [20] proposed Eq. (1.1) in the form of Schrödinger-Newton system to model a single particle moving in its own gravitational field.
As we know, standing wave solution of Eq. (1.1) corresponds to solution of the Choquard equation −∆u + u = I 2 * |u| 2 u in R 3 .
(1. 2) In detail, with a suitable scaling, the wave function ψ(x, t) = e −it u(x) is a solution of Eq. (1.1) once u is a solution of Eq. (1.2). Lieb demonstrated the seminal work on Eq. (1.2) in [16], in which he certified the existence and uniqueness (up to translations) of positive radial ground state solution by applying symmetrically decreasing rearrangement inequalities. After this, Lions [18] studied the same problem and further proved the existence of infinitely many radial solutions via the variational methods. From mathematical perspective, scholars prefer to study the general Choquard equation −∆u + W(x)u = γ (I α * G(u)) g(u) in R N , (1.3) where N ≥ 3, γ ∈ R + , I α is the Riesz potential of order α ∈ (0, N) defined for x ∈ R N \{0} by Γ is the Gamma function, * is convolution, W ∈ C(R N , R), g ∈ C(R, R) and G(u) = u 0 g(s)ds. To establish the variational framework for Choquard equations, we need the following celebrated Hardy-Littlewood-Sobolev inequality.
Thanks to (1.4), the integral R N (I α * |u| p )|u| p dx is well defined in H 1 (R N ) once p ∈ [2 α * , 2 * α ], where 2 * α := N+α N−2 and 2 α * := N+α N are usually called upper and lower critical exponents with respect to the Hardy-Littlewood-Sobolev inequality, respectively. It is easy to clarify that the critical terms R N (I α * |u| 2 * α )|u| 2 * α dx and R N (I α * |u| 2 α * )|u| 2 α * dx are invariant under the scaling actions σ N−2 2 u(σ·) and σ N 2 u(σ·) (σ > 0), respectively, and these two scaling actions served as group actions are noncompact on H 1 (R N ). Consequently, from the perspective of variational methods, the critical exponents 2 α * and 2 * α may provoke two kinds of lack of compactness. However, fortunately, similar to the Sobolev critical case studied in [3], these two kinds of loss of compactness can be recovered to some extent by using the extremal functions of the Hardy-Littlewood-Sobolev inequality.
In [21], Moroz and Van Schaftingen studied the case of Eq. (1.3) that W(x) ≡ 1, γ = 1 p and G(u) = |u| p (p > 1), they proved the existence, regularity, radially symmetry and decaying property at infinity of ground state solution when p ∈ (2 α * , 2 * α ). Meanwhile, based on the regularity of solutions, they established a Nehari-Pohožaev type identity and then showed the nonexistence of nontrivial solutions for Eq. (1.3) when p / ∈ (2 α * , 2 * α ). Afterwards, in [22], they extended the existence results in [21] to the case of Eq. (1.3) that g satisfies the so-called almost necessary conditions of Berestycki-Lions type. For the critical cases of Eq. (1.3), with the nonexistence result of [21] in hand, an increasing number of scholars devote to studying Eq. (1.3) with critical term and a noncritical perturbed term. We refer the interested readers to [4,9,14,30] for upper critical case, [23,26] for lower critical case and [15,25,31] for doubly critical case.
When it comes to the case W(x) ̸ ≡ const., we focus our attention on steep well potential of the form λV(x) + b, where λ > 0, b ∈ R and V ∈ C(R N , R) satisfies the following hypotheses: This type of potential was firstly introduced by Bartch and Wang in [2] to study the existence and multiplicity of nontrivial solutions for subcritical Schrödinger equations in the case of b > 0. Later, Ding and Szulkin further considered the case b = 0 in [8]. Since |Ω| < +∞, then −∆ possesses a sequence of positive Dirichlet eigenvalues µ 1 < µ 2 < · · · < µ n → +∞. Assuming b < 0 and b ̸ = −µ i for any i ∈ N + , Clapp and Ding [6], together with Tang [27], studied the existence and concentration of ground state solution for critical Schrödinger equation. Recently, the pre-existing results on Schrödinger equations have been extended to the Choquard equations, see e.g. [1,14,15,19] and the references therein.
It follows from (V 3 ) that the first eigenvalue λ 1 of −∆u , 2 , following the ideas in [5], they derived a ground state sign-changing solution by using minimization arguments in sign-changing Nehari manifold.
Motivated by the above works, in the present paper, we study the Choquard equation and the norm ∥ · ∥ λ = (·, ·) 1 2 λ for any λ > 0. Since V ≥ 0 in R N , it is easy to see that E λ → H 1 (R N ) and, for any s ∈ [2, 2 * ], there is some constant ν s > 0 such that, for all λ > 0, By (1.4) and (1.7), we deduce the energy functional J λ,µ of Eq. (1.6) belongs to Now we are prepared to state our main results.

Remark 1.3.
Similar to the proof of Theorem 1.1 in [14], by minimizing J λ,µ on the Nehari manifold we can demonstrate that Eq. (1.6) has a positive ground state solution v λ,µ for any λ, Thereby, in a standard way, we may deduce J ′ λ,µ (|u λ,µ |) = 0. Whereas, the strong maximum principle implies |u λ,µ | > 0 in R N , and the regular estimates for Choquard equations (see e.g. [21,22]) implies u λ,µ ∈ C(R N , R), thus u λ,µ has constant sign in R N , which contradicts with u ± λ,µ ̸ = 0. Furthermore, due to the presence of the perturbed term µ|u| p−2 u, the methods introduced in [11,32] to verify that the least energy of sign-changing solutions is less than twice the least energy of nontrivial solutions seem invalid here, we propose an open question whether J λ,µ (u λ,µ ) < 2J λ,µ (v λ,µ ). Remark 1.4. To our knowledge, there seem to be no results on (ground state) sign-changing solutions for Choquard equations with upper critical exponent, even on the bounded domain. Our present work extends and improves the existence results of sign-changing solutions verified in [7,10,11,28,33]. In [5], the authors studied the ground state sign-changing solutions for a class of critical Schrödinger equations is a bounded domain and λ ∈ (0, λ 1 ), with λ 1 denoting the first eigenvalue of −∆ on D. They proved that any sign-changing (PS) c sequence is relatively compact once c < c 0 + 1 N S N 2 , where c 0 is the least energy of nontrivial solutions. As a counterpart for the work in [5], Zhong and Tang studied a class of Choquard equations with critical Sobolev exponent in [33], where they showed the relative compactness of sign-changing (PS) c sequence with c less than the similar threshold. However, in this paper, due to the presence of the upper critical nonlocal term (I α * |u| 2 * α )|u| 2 * α −2 u in Eq. (1.6), the relative compactness of sign-changing (PS) c sequence with α cannot be deduced as expected, where S α is defined by (2.12) hereinafter. Also, it seems intractable to search for sign-changing (PS) c sequence such that c < 2+α 2(N+α) S (N+α)/(2+α) α for small µ > 0. Naturally, we attempt to construct a sign-changing (PS) c sequence with c < 2+α 2(N+α) S (N+α)/(2+α) α by assuming that µ > 0 is sufficiently large. Therefrom, by applying the properties of steep well potential λV, we can standardly prove the relative compactness of this type of sign-changing (PS) c sequence and then obtain ground state sign-changing solution.
We will give the proof of Theorem 1.2 in the forthcoming section. Throughout this paper, we use the following notations: (1) is a quantity tending to 0 as n → ∞ and |Ω| is the Lebesgue measure of Ω ⊂ R N .

Proof of Theorem 1.2
For the limiting problem of Eq. (1.6) as λ → +∞, namely Eq. (1.8), its energy functional is Due to (1.4) and Define the sign-changing Nehari manifolds Clearly, M λ,µ and M ∞,µ contain all of the sign-changing solutions of Eqs. (1.6) and (1.8), respectively. To search for ground state sign-changing solutions, we consider the following minimization problems: Before completing the proof of Theorem 1.2, we establish several preliminary lemmas.
Proof. From the definition of {u n }, there results Then there exists some C 2 > 0 independent of λ and µ such that lim sup n ∥u n ∥ λ ≤ C 2 . Naturally, {u n } is bounded in E λ . Hence, there exists some u ∈ E λ such that, up to subsequences, (2.20) Set v n = u n − u. Clearly, lim sup n ∥v n ∥ λ ≤ 2C 2 . We will show ∥v n ∥ λ n − → 0 up to a subsequence. Define We assert β = 0. Otherwise, β > 0. Due to (V 5 ), there exists some large R > 0 such that Then it follows from the Hölder and Sobolev inequalities that which contradicts β > 0. That is, our claim β = 0 is true. Then, thanks to [29,Lemma 1.21], v n → 0 in L s (R N ), ∀ s ∈ (2, 2 * ).
which contradicts c < m * . Thus u n → u in E λ up to a subsequence. This lemma is proved.
Based on the above preliminary lemmas, we shall complete the proof of main results below.