In this paper, we study the following upper critical fractional Kirchhoff-Choquard type equation

with prescribed L 2 -norm constraint

where s ∈ ( 0 , 1 ) , N ∈ ( 2 s , 4 s ) , θ ∈ ( 0 , N ) , a , b , c , μ > 0 , p ∈ ( N + θ N , N + θ + 2 s N ) , N + θ + 2 s N is the L 2 -critical exponent, N + θ N and 2 θ , s * : = N + θ N − 2 s are lower critical exponent and upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Furthermore, λ appears as an unknown Lagrange multiplier, V is an external potential vanishing at infinity, the function I θ : ℝ N → ℝ is called the Riesz potential and defined as follows,

with Γ representing the Gamma function. For convenience, we drop A θ in what follows. The symbol ( − Δ ) s is the fractional Laplace operator defined as

where the symbol P.V. means the Cauchy principal value of the singular integral and C ( N , s ) is a dimensional constant, precisely given by C ( N , s ) : = ( ∫ ℝ N ( ( 1 − cos ζ 1 ) / ( | ζ | N + 2 s ) ) d ζ ) − 1 . H s ( ℝ N ) is defined in detail below. The fractional Laplacian operator appears in diverse areas such as financial mathematics, optimization and minimal surfaces etc.

In recent years, significant attention has been paid to the following Choquard equation

which arises from several physical contexts. When N = 3 , θ = p = 2 , (1.3) turns to be the well-known Choquard-Pekar equation by S. Pekar [1] in 1954 which was firstly introduced as a model in quantum theory of a polaron at rest. In 1976, (1.3) was used by Choquard [2] to study steady states of the one component plasma approximation to Hartree-Fock theory. In recent years, the existence of solutions to (1.3) was demonstrated mathematically in [3]-[5] via variational approaches. Moroz and Schaftingen [6] proved the existence of the positive ground state solution to the Equation (1.3) with p > 1 , and established the regularity, positivity, monotonicity and decay asymptotics of the normalized ground state solutions. For N ≥ 3 , p = N + θ N − 2 and q ∈ ( 2 , 2 + 4 N ) , Li [7] considered the existence and orbital stability of the normalized ground states for the Choquard equation with a local perturbation μ | u | q − 2 u under the L 2 -norm constraint. Shang and Ma [8] considered the existence of positive solutions for the following Choquard equation

where N ≥ 3 and N + θ N < p < 2 θ * : = N + θ N − 2 . Shang and Ma established a set of existence results and characterized the asymptotic behavior as μ → 0 + by setting various assumptions on the parameter p . Meng only [9] considered the first case in (1.4) where p ∈ ( N + θ N , N + θ + 2 s N ) , and extended it to the fractional setting. Long, Li and Rong [10] proved normalized ground state solutions to the following Choquard equations with weakly attractive potential under the L 2 -norm constraint

where N ≥ 3 , p , q ∈ ( N + θ + 2 N , N + θ N − 2 ) and the trapping potential V .

On the other hand, extensive research efforts have also been focused on exploring the normalized ground state solutions to Kirchhoff equations. The Kirchhoff equation was first introduced by Kirchhoff [11] in 1883 as an extension of classical D’Alembert’s wave equation. For instance, Ye [12] analyzed the exclusion of dichotomy of the minimizing sequences for the related constrained minimization problem to prove the existence of solutions with L 2 -norm constraint for the following Kirchhoff equation

where N ≤ 3 , p ∈ ( 2 , 2 * ) , 2 * = 6 if N = 3 and 2 * = + ∞ if N = 1 , 2 . Especially, Ye obtained the sharp existence of global constraint minimizers for 2 < p ≤ 2 + 8 N . Ji et al. [13] developed a perturbed Pohozaev constraint approach to consider the Kirchhoff equation under L 2 -norm constraint as follows

where 2 < q < 14 3 < p ≤ 6 or 14 3 < q < p ≤ 6 , and proved several asymptotic results for the normalized ground state solutions obtained. Hu and Mao [14] partially extended the results of (1.6) under different assumptions on the exponents 2 < p < q < 6 . They have studied the existence and nonexistence of the normalized ground state solutions based on the methods of constrained minimization and concentration compactness. Chen and Huang [15] proved the existence and nonexistence of solutions to the fractional Kirchhoff equation with an external potential V and doubly critical exponent.

Recently, some researchers have dedicated their efforts to exploring the existence of normalized ground state solutions of Kirchhoff-Choquard equation. In [16], Liu established the threshold values for the existence and nonexistence of solutions to the Kirchhoff-Choquard equation. Furthermore, the asymptotic behaviors of the corresponding Lagrange multipliers and energies as c → 0 and c → ∞ were obtained. Zhu et al. [17] developed a new perturbed Pohozaev constraint approach to prove the existence of two normalized ground state solutions for the following Kirchhoff-Choquard equation under L 2 -norm constraint

where N ≥ 3 and N + θ N < q < N + 4 + θ N < p < 2 θ * or N + 4 + θ N < q < p < 2 θ * , and obtained the asymptotic behavior for the normalized ground state solutions as μ → 0 + . Liu and Sun [18] focused on the existence of normalized ground state solutions to the Kirchhoff-Choquard equation under L 2 -norm constraint as follows

where the potential V satisfies different conditions.

The key difficulty in combining the upper critical exponent with a vanishing potential in the normalized framework is the possible loss of compactness of minimizing sequences, which may exhibit dichotomy rather than convergence or vanishing. Our analysis, particularly in Lemma 3.10 and Theorem 1.5, is designed to exclude this dichotomy. Based on these results, we prove the existence of normalized solutions to the fractional Kirchhoff-Choquard Equation (1.1) with an HLS upper critical exponent and a potential, using a perturbed Pohozaev constraint and variational methods. Furthermore, we establish the asymptotic properties of the normalized ground states as μ → 0 + in the autonomous case.

under the L 2 -norm constraint

We now define several important types of spaces that will be essential in subsequent sections. To begin with, we define that |   ⋅   | p is the standard norm in Lebesgue space L p ( ℝ N ) for p ∈ [ 1 , ∞ ) . Subsequently, we define that

is the standard norm in Sobolev space

where 2 s * : = 2 N N − 2 s . Finally and most importantly, we introduce the definition and some useful facts of the fractional Sobolev space H s ( ℝ N ) . As usual, the fractional Sobolev space H s ( ℝ N ) is defined for any s ∈ ( 0 , 1 ) as

Note from [19] that

The space H s ( ℝ N ) is a Hilbert space endowed with the following inner product and norm

and

Now, to establish our main results, we introduce the following auxiliary function

and proceed to list the key assumptions on W ( x ) and V ( x ) .

(A1): V ( x ) ∈ C 1 ( ℝ N ) \ { 0 } , V ∞ : = lim | x | → ∞ V ( x ) = 0 , sup x ∈ ℝ N V ( x ) = 0 and there exists σ 1 ∈ [ 0 , 1 ) such that ∫ ℝ N | V | u 2 ≤ σ 1 ∫ ℝ N | ( − Δ ) s 2 u | 2 d x for any u ∈ H s ( ℝ N ) .

(A2): W ( x ) ∈ C 1 ( ℝ N ) , lim | x | → ∞ W ( x ) = 0 , V ( x ) + 1 s W ( x ) ≤ 0 , ∀ x ∈ ℝ N .

(A3): V ( x ) is a radial function, non-decreasing with respect to r = | x | .

Remark 1.1.Let

If | V | N 2 s < S , then there exists σ 1 : = S − 1 | V | N 2 s < 1 and

For a sufficiently small ε > 0 , the function V ( x ) = − ε 1 + | x | 2 s on ℝ N satisfies the above conditions. Now we recall the definition of normalized ground state solutions.

Definition 1.2.For any fixed c > 0 , we say that u 0 ∈ S c is a normalized ground state solution to(1.1) under the constraint(1.2) if

Prior to studying the nonautonomous Equation (1.1), we introduce the following fractional Kirchhoff-Choquard equation in the autonomous case, i.e. V ( x ) ≡ V ∞ = 0

under the L 2 -norm constraint ∫ ℝ N | u | 2 d x = c . Firstly, we state the existence and the asymptotic properties of the normalized ground state solutions in the autonomous case. This is accomplished in the following Theorem 1.3 and 1.4.

Theorem 1.3.Assume that N ∈ ( 2 s , 4 s ) and p ∈ ( N + θ N , N + θ + 2 s N ) . For any μ > 0 , there exists a constant c ∞ : = c ∞ ( μ ) > 0 such that for any c ∈ ( 0 , c ∞ ) , problem(1.10)under the L 2 -norm constraint S c possesses a normalized ground state solution u ˜ c ∞ , μ ∈ H s ( ℝ N ) such that E ∞ , μ ( u ˜ c ∞ , μ ) < 0 and λ c ∞ , μ < 0 .

Theorem 1.4.Assume that N ∈ ( 2 s , 4 s ) and p ∈ ( N + θ N , N + θ + 2 s N ) . For any fixed c ∈ ( 0 , c ∞ ) , let u c ∞ , μ ∈ S c be the corresponding normalized ground state solution for μ > 0 sufficiently small, then

These results from the autonomous case play a crucial role in addressing our main problem (1.1). Next, we state the existence of the normalized ground state solutions in the nonautonomous cases.

Theorem 1.5.Assume that N ∈ ( 2 s , 4 s ) , p ∈ ( N + θ N , N + θ + 2 s N ) and V satisfies(A1), (A2)and(A3). For any μ > 0 , there exists a constant c 0 = c 0 ( μ ) > 0 such that for any c ∈ ( 0 , c 0 ) , problem(1.1)under the L 2 -normconstraint S c possesses a normalized ground state solution u c , μ ∈ H s ( ℝ N ) such that E μ ( u c , μ ) < 0 and λ c , μ < 0 .

This paper is organized as follows. In Section 2, we give some preliminary results which will be used in proving main results. In Section 3, we study existence and the asymptotic properties of the normalized ground state solutions in the autonomous case and prove Theorem 1.3 and Theorem 1.4. In Section 4, we focus on the existence of the normalized ground state solutions in the nonautonomous case and prove Theorem 1.5.

Throughout this paper, we adopt the following notations.

B r ( y ) : = { x ∈ ℝ N : | x − y | < r } . C denotes any positive constants possibly different in different places.The symbol ⇀ denotes weak convergence and the symbol → denotes strong convergence. o n ( 1 ) denotes a real sequence tending to 0 as n → ∞ . H − s ( ℝ N ) is the dual space of H s ( ℝ N ) .

In Section 2, we give several preliminary results which are significant for the subsequent development of our work. Especially, for convenience, we define

Then, according to (1.9), we have

and

Firstly, let us recall the well-known Hardy-Littlewood-Sobolev inequality and fractional Gagliardo-Nirenberg inequality.

Lemma 2.1. ([20]) Let r , t > 1 , θ ∈ ( 0 , N ) satisfy 1 r + 1 t = N + θ N , f ∈ L r ( ℝ N ) and g ∈ L t ( ℝ N ) . Then there exists a sharp constant C ( N , θ , r , t ) > 0 , independent of f and g , such that

Lemma 2.2. ([9]) Let N > 2 s and N + θ N < p < 2 θ , s * . Then there exists a constant C ( N , θ , s , p ) > 0 such that

Let S θ be best constant

it is well-known that S θ is achieved if and only if u is the form

for some x 0 ∈ ℝ N , C > 0 and ε > 0 see [21].

We give classical Brezis-Lieb lemma for the nonlinear local term.

Lemma 2.3.Let Ω ⊆ ℝ N be a domain, p ∈ [ 1 , ∞ ) and { u n } n ∈ ℕ be a bounded sequence in L r ( Ω ) . If u n → u a.e.on Ω as n → ∞ , then for every p ∈ [ 1 , r ] ,

We give Brezis-Lieb lemma for the convolution term of the functional.

Lemma 2.4. ([6]) Let N > 2 s , p ∈ [ 1 , ∞ ) and { u n } n ∈ ℕ be a bounded sequence in L 2 N p N + θ ( ℝ N ) . If u n → u a.e.on ℝ N as n → ∞ , then

Lemma 2.5. ([22]) Let R > 0 and p ∈ [ 2 , 2 s * ) . If { u n } is bounded in H s ( ℝ N ) and

then u n → 0 in L t ( ℝ N ) as n → ∞ for any t ∈ ( 2 , 2 s * ) .

Lemma 2.6.Assume that V satisfies(A1), (A2) and (A3).If { u n } ⊂ H s ( ℝ N ) satisfies u n ⇀ u in H s ( ℝ N ) and u n ( x ) → u ( x ) a.e.in ℝ N for some u ∈ H s ( ℝ N ) as n → ∞ ,there holds that

Proof:Since u n ⇀ u as n → ∞ in H s ( ℝ N ) , in view of Lemma 2.3, it implies that

and

Combining (2.5) with (2.6), we have

Hence, we can get that

Next, we let f n : = | V | 1 / 2 u n . It follows (A₁) implies that { f n } is bounded in L 2 ( ℝ N ) . Since u n → u a.e. as n → ∞ , we have f n → | V | 1 / 2 u = : f a.e. as n → ∞ . Applying Lemma 2.3 with p = 2 and r = 2 to the sequence { f n } , we deduce that

Substituting back f n = | V | 1 / 2 u n and f = | V | 1 / 2 u , and noting that V ≤ 0 (so | V | = − V ), we obtain

It follows Lemma 2.4, (2.8) and (2.7) that

Hence, the proof is complete. □

Now, we introduce the Pohozaev manifold

with

and it is easy to prove

is a smooth manifold.

Lemma 2.7.Assume that V satisfies(A1), (A2) and (A3). Suppose that u ∈ H s ( ℝ N ) is a weak solution of problem(1.1) under the L 2 -norm constraint S c , then

.

Proof: Since u is a solution of problem (1.1) under the L 2 -norm constraint S c , by [6] and [23], we know that u satisfies the Pohozaev identity

where we have used the fact that

Thus, one has

which means that

. □

For u ∈ S c and t ∈ ℝ , we set

then t ⋆ u ∈ S c . We define the map,

Then we easily obtain that

As a result, we get the following conclusion.

Lemma 2.8.Let u ∈ S c and V satisfies(A1), (A2) and (A3).Then t ∈ ℝ is critical point of I μ ( t ) ifand only if

.

Lemma 2.9.Assume that V satisfies(A1), (A2) and (A3). The function c → e ( c ) is continuous for any c ∈ ( 0 , c 0 ) .

Proof:For any c ∈ ( c , c 0 ) and { c n } ⊂ ( 0 , c 0 ) such that c n → c as n → ∞ . From the definition of e ( c n ) and Lemma 2.12, for every n there exists u n ∈ U ( c n ) such that

We set v n : = c c n u n , and hence { v n } ⊂ S c . Since E μ ( u ) < 0 , in view of (2.16) and Lemma 2.11, one has

which implies that v n ∈ U ( c ) . Thus, we have

From { u n } ⊂ U ( c n ) , Gagliardo-Nirenberg inequality (2.3) and (2.4), we get the boundedness of A ( u ) and B ( u ) . Then, combining (2.12) with (2.13), we have

On the other hand, let { w n } ⊂ U ( c ) be a minimizing sequence for e ( c ) with E μ ( w n ) < 0 . Set z n : = c n c w n , then { z n } ⊂ U ( c n ) . Therefore, we obtain that