Normalized solutions for the double-phase problem with nonlocal reaction

: In this article, we consider the double-phase problem with nonlocal reaction. For the autonomous case, we introduce the methods of the Pohozaev manifold, Hardy-Littlewood Sobolev subcritical approximation, adding the mass term to prove the existence and nonexistence of normalized solutions to this problem. For the nonautonomous case, we show the existence of normalized solutions to the double-phase problem by using the Pohozaev restrict method and describing the relationship between the energy of this problem and its limit problem. Moreover, we study the existence of normalized solutions to the double-phase problem involving double Hardy-Littlewood-Sobolev critical exponents


Introduction and preliminary results
In this article, we are concerned with the solutions of the following double-phase problem with the normalized constraint , the mass constant > c 0, the frequency ∈ λ is unknown and appears as Lagrange multiplier, and I α is the Riesz potential of order ( ) The study of equation (1.1) is motivated by recent fundamental progress in the mathematical analysis of many nonlinear patterns with unbalanced growth and nonlocal reaction.The main novelty of (1.1) is the combination of a double-phase operator and two nonlocal Choquard reaction terms.Since equation (1.1) is closely concerned with unbalanced double-phase problems and nonlocal Choquard problems.We briefly introduce in what follows the related background and applications and recall some pioneering contributions in these fields.Problem (1.1) combines an interesting phenomenon that the operator involved in (1.1) is the socalled double-phase operator whose behavior switches between two different elliptic situations, which generates a double-phase associated energy.Originally, the idea to treat such operators comes from Zhikov [61] who introduced such classes to provide models of strongly anisotropic materials, see also the monograph of Zhikov et al. [62].We refer to the remarkable works initiated by Marcellini [37,38], where the author investigated the regularity and existence of solutions of elliptic equations with unbalanced growth conditions.We also mention the recent paper by Mingione and Rădulescu [39], which is a comprehensive overview of the recent developments concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators.
The double-phase problem (1.1) is motivated by numerous models arising in mathematical physics.For instance, we can refer to the following Born-Infeld equation [11] that appears in electromagnetism: 2 1   2 Indeed, by the Taylor formula, we have By taking 2 and adopting the first-order approximation, we obtain the double problem for = p 4 and = q 2.Moreover, the nth-order approximation problem is driven by the multiphase differential operator We also refer to the following fourth-order relativistic operator which describes large classes of phenomena arising in relativistic quantum mechanics.Again, by Taylor's formula, we have This shows that the fourth-order relativistic operator can be approximated by the following autonomous double-phase operator For more details on the physical background and other applications, we refer the readers to Bahrouni et al. [6] (for phenomena associated with transonic flows) and to Benci et al. [9] (for models arising in quantum physics).
In the past few decades, the double-phase problem has been the subject of extensive mathematical studies.Using various variational and topological arguments, many authors studied the existence and multiplicity results of nontrivial solutions, ground state solutions, nodal solutions, and some qualitative properties of solutions, respectively, such as [21,43,46] in case of bounded domains.In this classical setting, we recall the seminal papers by Ni and Wei [44], Li and Nirenberg [29], del Pino and Felmer [18], del Pino et al. [19], and Ambrosetti and Malchiodi [5].The regularity, existence of solutions, and multiplicity of the double-phase problem on the whole space can be found in [3,24,58].The qualitative and asymptotic analysis of solutions for some related elliptic problems can refer to [13,14,28,47,60].
The second interesting phenomenon in equation (1.1) is the appearance of Choquard reaction terms, which generate the nonlocal characteristic.The nonlocal problem goes back to the description of the quantum theory of a polaron at rest by Pekar in [48] and the modeling of an electron trapped in its own hole in the work of Choquard, as a certain approximation to Hartree-Fock theory of one-component plasma [30].In some particular cases, the Choquard equation is also known as the Schrödinger-Newton equation, which was introduced by Penrose in his discussion on the selfgravitational collapse of a quantum mechanical wave function [49].The existence and qualitative properties of ground state solutions of Choquard equation have been widely studied in the last decades.See, for example, [32,35,[40][41][42].
Recently, physicists are often interested in the existence of normalized solutions.Indeed, prescribed mass appears in nonlinear optics and the theory of Bose-Einstein condensates, see [20,36] and the references, therein.In particular, when = = p q 2, ( ) ≡ V x 0, problem (1.1) is reduced to the following well known Cho- quard type equation: Cazenave and Lions in [16] proved the existence and stability of normalized solutions for (1.2) with = N 3 and = − α N 1 by considering the following minimizing problem: where In such case, we remark that (1.2) is a mass-subcritical minimizing problem, and essentially all results of (1.2) can be extended to the general case where ≥ N 3 and 2) becomes mass-supercritical and = −∞ ϑ c has no minimizers.Luo in [34] proved the existence of unstable ground state normalized solution for (1.2) with ≥ N 3 and < < − α N 0 2by considering a constrained minimizing problem on a suitable submanifold of S ˜. c For the following general Choquard equation: Here, ( ) and Ye in [27] studied the existence of normalized solutions for (1.3) with by using a minimax theorem developed by Bellazzini et al. in [8].Moreover, Ye in [56] obtained the existence of ground state normalized solution for (1. 3) with a trapping potential and = + + m .

N α N
2 In [15], Cao et al. focused on the existence of normalized solutions to the following equation with van der Waals type potentials: (1.4) under the normalized constraint S ˜. c Compared with the well-studied case = α β, the solution set of (1.4) with different widths of two body potentials ≠ α β is much richer.Under different assumptions on c, α and β, they proved the existence, multiplicity, and asymptotic behavior of solutions to (1.4)In addition, the stability of the corresponding standing waves for the related time-dependent problem is discussed.In particular, when = − β N 4 and ≥ N 5, Jia and Luo in [25] showed some existence, nonexistence, multiplicity, and asymptotic results of normalized solutions to (1.4).These results are a continuation of works in [15].Moreover, Yao et al. in [55] studied the following Choquard equations with lower critical exponent and a local perturbation They proved several nonexistence and existence results by introducing some new arguments.In particular, they first considered the existence of normalized solutions to (1.5) involving double critical exponents and Normalized solutions for the double-phase problem with nonlocal reaction  3 described some qualitative properties of the solutions with prescribed mass and of the associated Lagrange multipliers λ.
On the other hand, when < = < p q N 1 , ( ) ≡ V x 0, the Choquard term is replaced by | | − u u l 2 , there are few results on the following p-Laplacian equation: (1.6) In particular, when Li and Yan [26] obtained the existence of normalized ground state solutions.In [23], Gu et al. proved the existence of normalized ground state solutions with a trapping potential for (1.6) in case of = l p ˆ. Recently, Zhang and Zhang [60] considered the following p-Laplacian equation with a L p -norm constraint: Assume that g is odd and L p -supercritical.When < q p ând > μ 0, using Schwarz rearrangement and Ekeland variational principle, they proved the existence of positive radial ground states for suitable μ.When = q p ˆand > μ 0 or ≤ q p ˆand ≤ μ 0, with an additional condition of g, they proved a positive radial ground state if μ lies in a suitable range by the Schwarz rearrangement and minimax theorems.Via a fountain theorem type argument, with suitable ∈ μ , they showed the existence of infinitely many radial solutions for any ≥ N 2 and the existence of infinitely many nonradial sign-changing solutions for = N 4 or ≥ N 6.In addition, Baldelli and Yang in [7] were con- cerned with the existence of normalized solutions to the following ( ) q 2, -Laplacian equation in all possible cases according to the value of p with respect to the L 2 -critical exponent In the L 2 -subcritical case, they studied a global minimization problem and obtained a ground state solution.
While in the L 2 -critical case, they proved several nonexistence results, extended also in the L q -critical case: For the L 2 -supercritical case, they derived a ground state and infinitely many radial solutions.
Inspired by the aforementioned literature, we want to study the normalized solutions of the double-phase problem with nonlocal reaction (1.1).
The features of equation (1.1) are the following: (i) The presence of several differential operators with different growth, which generates a double-phase associated energy.(ii) The equation combines the multiple effects generated by two nonlocal terms and a variable potential.(iii) Due to the unboundedness of the domain, the Palais-Smale sequence does not have the compactness property.
Throughout this article, for any is the usual Sobolev space endowed with the norm For equation (1.1), we introduce the working space E endowed with the norm In addition, for given { } ∈ u E\ 0 and > t 0, we define the scaling function: which remains in E and preserves the L p norm when > .
To study equation (1.1) variationally, we give the following Hardy-Littlewood-Sobolev inequality and the Gagliardo-Nirenberg inequality. (1.9) In this case, there is equality in (1.9) if and only if Then for ( ) Thus, power p α is called Hardy-Littlewood-Sobolev lower critical exponent and power p α is called Hardy- Littlewood-Sobolev upper critical exponent.Combining (1.9), we denote Normalized solutions for the double-phase problem with nonlocal reaction  5 There exists a constant > S 0 such that, for any ( ) The space E is embedded continuously into ( ) Lemma 1.6.[2,45] The following results hold: and ≤ < < p m q 1 *.Then there exists a sharp constant

Nq m p m Nq p N q
Based on the aforementioned facts, we shall consider the existence and nonexistence of normalized solutions to equation (1.1).It is well known that the normalized solution of equation (1.1) can be obtained by looking for a critical point of the following functional , where the potential conditions that ensure that the potential term is well defined will be given in the following argument.In addition, combining Lemmas 1.1 and 1.6, we find that (1.15) and (1.16) for some positive constants C K , Then on the basis of (1.15) In addition, when the functional I V l , is bounded from below on S c , we shall give some results about the minimization problem In order to search for critical points of I V l , restricted to S c , we shall use the Pohozaev manifold c V l , as a natural constraint of I V l , that contains all the critical points of I V l , restricted to S c , where First, we study the existence and nonexistence of normalized solutions to equation (1.1) in case of ( ) ≡ V x 0. Hence, inspired by [17] and [50], we define the fibering map given by Obviously, for any ∈ u S c , the dilated function u t belongs to the constraint manifold c l 0, if and only if ∈ t is a critical value of the fibering map Thus, it is natural to split c l 0, into three parts corresponding to local minima, local maxima, and points of inflection.From [52], we define Now we state our main results about the autonomous problem as follows.
Theorem Normalized solutions for the double-phase problem with nonlocal reaction  7 Remark 1.8.To prove that a c is achieved, we need to introduce the compactness principle in [33].However, compared with [55], due to the influence of double-phase operator and Hardy-Littlewood-Sobolev lower critical exponent, it is difficult to make an estimate on the properties of a c in advance.To overcome this difficulty, we shall give the description between S α and c in detail.
Next, we make most of Pohozaev equality, Hardy-Littlewood-Sobolev inequality, and Gagliardo-Nirenberg inequality (1.16) about the nonlocal Choquard term, we give the following nonexistence result.Theorem 1.9.Assume that > γ 0, = l p ¯, and q lγ l q l γ l q p lγ l q lγ l q 2 .
α α p q q N l α , , α lγ l q lγ l q p p α ql γ l lγ l q lγ l q p p α ql γ l q p p α lγ l q ql γ l Remark 1.11.To show the existence of normalized solutions to equation (1.1) with ( ) ≡ V x 0 in L p -supercritical case, we use the minimax method introduced in [17] to construct a Palais-Smale sequence.But because of the double-phase operator and Hardy-Littlewood-Sobolev low critical exponent, we need to rely on the Pohozaev manifold method to overcome the lack of compactness.Theorem 1.12.Assume that > μ 0, = l p .then there exists > μ ˜0 large enough such that for every > μ μ ˜, equation (1.1) has a ground state u for some > λ 0, which is positive, radially symmetric, and nonincreasing in N , where c ˇis introduced in Theorem 1.10 and Remark 1.13.In Theorem 1.12, in order to overcome the technical difficulties caused by the emergence of double Hardy-Littlewood-Sobolev critical terms, we introduce the new methods of adding mass term and the Hardy-Littlewood-Sobolev subcritical approximation to prove that m c p 0, α is achieved.
From the analysis of Theorems 1.7, 1.10 and 1.12, we in addition obtain the following properties of a c and m .c l 0, Theorem 1.14.The following results hold: (i) Suppose that the assumptions of Theorem 1.7 hold.Then the mapping ↦ c a c is a continuous and strictly decreasing mapping.
(ii) Suppose that the assumptions of Theorems 1.10 and 1.12 hold.Then the mapping ↦ c m c l 0, is continuous and strictly decreasing.
Secondly, we shall study the existence of normalized solutions to equation (1.1) in case of ( ) ≢ V x 0. To be more precise, ( ) V x satisfies the following conditions: x x N Moreover, there exist ¯2 q q q q q q q 2 such that for all ∈ u E, x Moreover, there exist such that for all ∈ u E, Moreover, there exist and such that for all ∈ u E, Normalized solutions for the double-phase problem with nonlocal reaction  9 W x 0 a.e. on .

N
Based on Theorems 1.10, 1.12, and 1.14, we treat equation (1.1) in case of ( ) ≡ V x 0 as the limit problem of equation (1.1) in case of ( ) ≢ V x 0 and give the following results.
Theorem 1.15.Assume that > γ 0, < ≤ p l p ¯α, and *, ** 0 such that for every 1) has a positive ground state u with some > λ 0, where μ ˆis introduced in Lemma 6.5, c ć ˇ, , and μ ˜are introduced in Theorems 1.10 and 1.12, respectively, ( , , and Remark 1.16.To our best knowledge, it seems to be the first work on the existence of normalized solutions for the double-phase problem with nonlocal critical reaction, potential, and mass supercritical perturbation.The appearance of different operators with different growth and potential term affect the geometry of this problem.Moreover, since is not compact.We shall use the monotonicity of energy to limit equation and describe the relationship between the energy of equation (1.1), and its limit equation in detail to overcome the lack of compactness and prove that m c V l , is achieved.This article is organized as follows.In Section 2, we study the existence of normalized solutions to (1.1) in case of ( ) ≡ V x 0 with L p -subcritical perturbation.In Section 3, we show the nonexistence of normalized solutions to (1.1) in case of ( ) ≡ V x 0 with L p -critical perturbation.Section 4 is devoted to proving the existence of normalized solutions to (1.1) in case of ( ) ≡ V x 0 with L p -supercritical perturbation.In Section 5, we give some properties of a c and m .c l 0, In Section 6, based on Sections 4 and 5, we consider the existence of normalized solutions to (1.1) in case of ( ) ≢ V x 0 with L p -supercritical perturbation.

The low critical leading term with focusing L p -subcritical perturbation
In this section, we consider the existence of normalized solutions to (1.1) in case of < < p l p α and ( ) ≡ V x 0. First, we give the following lemmas, which are necessary preparation.
In fact, for any ∈ u E, we define for > s 0 small enough.This implies that < a 0.
c By combining (2.1), we obtain that This is impossible and < < a τ a τ a .iii By using (1.10) and (1.11), we choose v such that for > s 0. Then ∈ u S s c for all > s 0. By direct calculation, we derive that for s small enough.Therefore, ( ) iii holds.
Then the sequence { } u n is relatively compact in E.
Proof.By Lemma 2.1(i), we know that { } u n is bounded in E. We claim that there exists > δ 0 such that n p , and by using By combining the Brezis-Lieb lemma for nonlocal nonlinearities in [40] and Lemma 2.1(ii), we obtain , namely, a c is attained by the real-valued positive and radially symmetric nonincreasing function ∈ u S ˜. c In addition, since u ˜is a critical point of I l 0, restricted to S c , there exists a Lagrange multiplier

The low critical leading term with focusing L p -critical perturbation
In this section, we mainly show the nonexistence of normalized solutions to (1.1) in case of = l p ¯and ( ) ≡ V x 0 under the suitable conditions.To be more precise, we give proof of Theorem 1.9.
Proof of Theorem 1.9.Assume that Let u be a solution to equation (1.1).By the Pohozeav identity, we know that ∈ u .c p 0, ¯Then from (1.16), we deduce that solutions for the double-phase problem with nonlocal reaction  13 This means that

This is
impossible.Then the proof of Theorem 1.9 is completed.□ Remark 3.1.From Theorem 1.9, if we let = l p ˜and ( ) ≡ V x 0, then letting u be a solution to equation (1.1) and using the Pohozaev identity and (1.15), we obtain that , which is a contradiction.To sum up, suppose that , ˜, 2 ˜1 p then equation (1.1) has no solution for any ∈ λ .

The low critical leading term with focusing L p -supercritical perturbation
In this section, we consider the existence of normalized solutions to (1.1) in case of < ≤ p l p ¯α and ( ) ≡ V x 0. We shall restrict critical points of I l 0, to a natural constraint manifold c l 0, , on which I l 0, is bounded below.
α α p q q N l α , , α lγ l q lγ l q p p α ql γ l lγ l q lγ l q p p α ql γ l q p p α lγ l q ql γ l Hence, we complete the proof of Lemma Proof.On the one hand, since ( ) ⊂ c l r c l 0, 0, , we find that On the other hand, we need to show that It follows from Lemma 4.1 that Fix ∈ u S c and let ∈ u S ˜cr be the Schwartz rearrangemnet of | | u , where r Then from (2.3), for all > t 0, we derive that Obviously, ( ) ( ) ( ) ( ) ˜, which along with (4.5) yields that To sum up, in view of Lemma 4.2, Hence, the proof of Lemma 4.

H uH
If Θ is contained in a connected component of ( ) Based on Lemma 4.3, we obtain that Then applying Lemma 4.4, we easily obtain the following lemma.

6) that
Normalized solutions for the double-phase problem with nonlocal reaction  17 Similar to the proof of Lemma 2.2, by Brezis-Lieb Lemma in [12], we have Moreover, Hence, in view of ( ) ( ) , we see that In addition, we observe that Combining the fact that , and there eixsts a unique ( ] ∈ t 0, 1 such that ( ) = P u ˜0.
Then we know that m c l 0, is attained by the real-valued positive and radially symmetric nonincreasing function.
Moreover, it follows from (4.2) that This means that q q N l α , , q lγ l q l γ l q p lγ l q lγ l q Thus the proof of Theorem 1. 10 is a radial nonincreasing function, then one has is the area of the unit sphere in .
a.e. on .N Furthermore, by the Lagrange multipliers rule, there exists ∈ λ n such that for every ∈ ψ E, Combining Lemma 1.1 and the Sobolev embedding theorem, we see that { } λ n is bounded.Thus, there exists This means that ≥ λ 0. Now we claim that ≠ λ 0. Otherwise, we obtain that Moreover, we have as → ∞ n , by the Hölder inequality, we derive that {| | } u n l n is bounded in ( ) .
1, , by the Young inequality, the Hölder inequality and Lemma 4.9 with = − ν Np N p , there exists a constant > C 0 independent of n such that By Lemma 1.1 and the Lebesgue-dominated convergence theorem, By combining this with (4.7), we deduce that This means that u is a weak solution of Now we show that ≠ u 0. Otherwise, by using ( ) = P u 0 l n 0, n , (1.12) and the Young inequality, (4.9) By using Lemma 4.7, (4.9), and the fact that (4.10) In addition, by using (1.10), we let , where S α is achieved by We note that From Lemma 4.1, there exists a unique constant By direct calculation, it follows from (4.11) that there exists > μ ˜0 large enough such that when > μ μ ˜, Then combining (4.10), we derive that However, since one has This is a contradiction.So ≠ u 0.

Next we show that
, we have and By combining the fact that ( ) = P u 0 p 0, α , we easily know that ≤ t 1. ω In addition, u satisfies the following equality Then we deduce that Normalized solutions for the double-phase problem with nonlocal reaction  23 On the other hand, similar to (4.13), we also have Therefore, in view of (4.14), one infers that Here, using the condition Hence, the proof of Lemma 4.10 is completed.□ Proof of Theorem 1.12.In view of Lemma 4.10, we know that w is a ground state solution to (1.1) for some > λ 0. The proof of Theorem 1.12 is completed.□ 5 Qualitative properties of the mappings a c and m c l 0, In this section, we discuss the continuity and monotonicity of a c and m .

The nonautonomous problem
In this section, we consider the existence of normalized solutions to equation (1.1) in case of < ≤ p l p ¯α and ( ) ≢ V x 0. From the condition ( ) V 1 , we know that equation (1.1) in case of ( ) ≡ V x 0 can be the limit problem to equation (1.1) in case of ( ) ≢ V x 0. Then the existence of normalized solutions and properties of energy m c l 0, to equation (1.1) in the autonomous case play the crucial role in this section, which will be explained later in the proof.
The following lemma helps us to show that I V l , is bounded away from 0 on .
, , 2 1 l lγ l q 1 2 (6.1) , , in view of ( ) ( ) p p q q q q p p q q q q q p q α l l p p q q N l α q lγ l γ Since < q lγ 2 l , we know that (6.1) holds.Hence, we complete the proof of Lemma 6.1.
, , by using ( ) V 1 , ( ) V 2 , we infer that p p q q q p p q q q q q p q α l l V l p p q q p q α α p p α l l p p q q α α p p p p p q q q p p q q q α α p p p q q q q N l α l γ α α p p p , into the disjoint union Now we give some properties of .
Normalized solutions for the double-phase problem with nonlocal reaction  29 u u V l p p q q q q q q q p q q p q p p q q q q q q q , 3 5 and we complete the proof of Lemma 6.3.□ Lemma 6.4.Assume that > γ μ , 0, < ≤ p l p ¯α, and ( ) V 1 , ( ) V 2 , ( ) V 3 hold.For any ∈ u S c , the function Ψ ¯u has a unique critical point s .
u That is, there exists a unique and Ψ ¯u is strictly decreasing and concave on ( ) +∞ s , .u ps p p q q δ s q q s p qδ s s q q qδ s s q s α l l ps p p q q q δ s q q s α l l u ps p p q q δ s q q s p qδ s s q s α l l q qδ s s q q qδ s s q ps p p q q q q δ s q q s α l l Now based on (6.4) and (6.7), we let It is clear that f 1 and f 2 have a unique zero point.Then we suppose that ( ) = f s ¯0 and ( ) = f s ¯0.Moreover, we find that ( ) > f s 0 Then it follows from (6.4) that Ψ ¯u is strictly increasing on ( ) −∞ s , ¯. , namely, , and ( ) V 3 , we deduce that ps p p q q q q δ s q q s α l l ps p p q q q q q q δ s q q 2 1 5 ¯2 6 N l α l q lγ l q 2 2 , , 2 α lγ l q p p α lγ l q ql γ l and c* is introduced in Lemma 6.2, then there exists > μ ˆ0 large enough such that when > μ μ ˆstaisfying and A ¯k is a closure of A .V l p p q q p q α α p p α l l p p q q α α p p p N l α q lγ l γ p p q q q q q q q p q q p q q q α l l p p q q q q q p q α l l N N N N (6.9) By combining (6.8) and (6.9), we obtain that p p q q q q q q q p q α l l p p q q q q q q q p q α l l V l c V l , , We shall divide the proof into three steps.
Step 1. we show that there eixsts a couple ( ) ∈ × , we introduce the minimax class with the associated minimax level  c V l V l V l p p q q q p q q α α p p p q n n p p q n q q n p q n q α α n p n p

0
Without loss of generality, we may assume that < < c c 0 ¯.Then Normalized solutions for the double-phase problem with nonlocal reaction  11 Hence, ( ) ii holds.( )

2 )
By assuming by contradiction and by Lemma I.1 in[33], we deduce that → u 0 n in ( ) L ν N for < < p ν p*.It follows from(1.11)and Lemma 1.1 that

2 by
Lemma 2.1(iii).Hence, the proof of Theorem 1.7 is completed.□ 3 is completed.□ Next, by applying the implicit function theorem on the 1 function × ϕ ϕ t u is an isomorphism, where T S u c r denotes the tangent space to S c r in u.Moreover, similar to proof of Lemmas 3.15 and 3.16 in [17], we have the following lemma.Lemma 4.4.Assume that > γ μ , 0, < ≤ p l p ¯α, and ( ) ≡ V x 0. Then the following results hold: (i) ( )[ ] ( Suppose that is a homotopy-stable family of compact subsets of S c r with closed boundary Θ and set restricted to S c r at level e . will establish the existence of a Palais-Smale sequence { } ( ) Repeating the same argument as proving ≠ u 0, we obtain that = b 0. Therefore, → u u n strongly in E. So the proof of Lemma 4.6 is completed.□ Proof of Theorem 1.10.In view of Lemmas 4.2 and 4.5, there exists a bounded Palais-Smale sequence { which along with Lemma 4.3 yields that u is a radially symmetric ground state solution of (1.1) for some > λ 0. Let u ˜be the Schwartz symmetrization rearrangement of | | u .Similar to the proof of Theorem 1.7, we have ( ) ≤ P u ˜0 l 0,

,
it follows from (4.8) that Normalized solutions for the double-phase problem with nonlocal reaction  21

we
find that for μ large enough, (4.15) holds.Hence, there exists > μ ˜0 such that when > μ μ ˜, (4.15) holds.In the last, since the infimum m c p 0, α is achieved by w, there exist λ and ξ such that

c l 0 ,
Next we give the details of proof in the following argument.

0
Without loss of generality, we may assume that < < − By combining this with the continuity of a c , we obtain that a c is strict decreasing.( ) ii Similar to the proof of ( ) i , we easily know that m c l 0, is continuous with respect to c. Now we show that ↦ c m c l 0, is strictly decreasing.Based on Theorems 1.10 and 1.12, let < < < +∞ c

ω
Then by simple calculation, one obtains is strictly decreasing.So we complete the proof of Theorem 1.14.□

1 2
, and (1.16), one has Normalized solutions for the double-phase problem with nonlocal reaction  27 So we complete the proof of Lemma 6.2.□Next, similar to the aforementioned we consider the decomposition of c V l

u
know that Ψ ¯u is strictly concave on ( ) +∞ s , .In the last, we shall show that the map ∈ ↦ ∈

8 )
Normalized solutions for the double-phase problem with nonlocal reaction  33On the other hand, since ∈ u

ζ 0 .
So the proof of Theorem 6.7 is completed.□ Proof of Theorem 1.15.To show that there exists ( )

2 1 ,
Then it is clear that I ˜V l , is of class .Inspired by[50,51], denoting by V l d in Γ.Then m ˜cV l , is a real number.Next, for any ( ) and (1.16), we know that ≔ 1 , 4.2.□Normalizedsolutions for the double-phase problem with nonlocal reaction  15