Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 56, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.56 NORMALIZED SOLUTIONS FOR SOBOLEV CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS YUXIN LI, XIAOJUN CHANG, ZHAOSHENG FENG Abstract. We study the Sobolev critical Schrödinger-Bopp-Podolsky system −∆u+ φu = λu+ µ|u|p−2u+ |u|4u in R3, −∆φ+ ∆2φ = 4πu2 in R3, under the mass constraint ∫ R3 u2 dx = c for some prescribed c > 0, where 2 < p < 8/3, µ > 0 is a parameter, and λ ∈ R is a Lagrange multiplier. By developing a constraint minimizing approach, we show that the above system admits a local minimizer. Furthermore, we establish the existence of normalized ground state solutions. 1. Introduction We consider the Schrödinger-Bopp-Podolsky system −∆u+ φu = λu+ µ|u|p−2u+ |u|4u in R3, −∆φ+ ∆2φ = 4πu2 in R3, (1.1) where u, φ : R3 → R, µ > 0, λ ∈ R and 2 < p < 8/3. System (1.1) was suggested as a model to describe solitary waves for nonlinear Schrödinger equation coupled with an electromagnetic field in the Bopp-Podolsky electromagnetic theory [11, 38]. The functions u and φ denote the modulus of the wave function and the electrostatic potential, respectively. The Bopp-Podolsky theory is a second-order gauge theory of the electromagnetic field, which was developed by Bopp [11] and Podolsky [38] independently to solve the alleged infinity problem in classical Maxwell theory. For more physical applications, we refer the reader to [9, 12, 13, 16, 17, 21, 23, 30] and the references therein. In the recent decades, considerable attention has been given to the Schrödinger- Bopp-Podolsky system from quite a few scientific fields. Siciliano-D’Avenia [22] studied a Schrödinger-Bopp-Podolsky system with Sobolev subcritical growth, −∆u+ ωu+ q2φu = |u|p−2u in R3, −∆φ+ a2∆2φ = 4πu2 in R3, (1.2) 2020 Mathematics Subject Classification. 35K92, 35B44, 35B40, 35R02. Key words and phrases. Normalized solutions; Schrödinger-Bopp-Podolsky system; Lagrange multiplier; ground state; variational method. ©2023. This work is licensed under a CC BY 4.0 license. Submitted May 28, 2023. Published September 5, 2023. 1 2 Y. X. LI, X. J. CHANG, Z. FENG EJDE-2023/56 where a > 0, ω > 0, q 6= 0, and p ∈ (2, 3]. They obtained the existence and nonex- istence results depending on the various ranges of p and q and showed that, in the radial case, those solutions tend to the solutions of the classical Schrödinger-Poisson equation as a→ 0. Silva-Siciliano [41] proved that the system has no solutions for large q and has two radial solutions for small q. They also presented qualitative properties about the energy level of the solutions and a variational characterization of these extremal values of q. Wang-Chen-Liu [44] established the existence, mul- tiplicity and asymptotic behavior of solutions for the Schrödinger-Bopp-Podolsky system with general nonlinearities. Figueiredo-Siciliano [24] proved the existence and multiplicity of solutions for the Schrödinger-Bopp-Podolsky system under pos- itive potentials. Chen-Tang [19] studied a critical Schrödinger-Bopp-Podolsky system with a sub- critical perturbation, −∆u+ V (x)u+ φu = µf(u) + u5 in R3, −∆φ+ a2∆2φ = 4πu2 in R3, (1.3) where a > 0, V , and f are continuous functions, and ∫ t 0 f(s) ds ≥ tp for p ∈ (2, 6) and t ≥ 0. They showed that system (1.3) admits ground state solutions under certain conditions of V and f . Using different variational techniques, Li- Pucci-Tang [31] obtained the existence of a nontrivial ground state solution for (1.3) when f(u) = |u|p−1u and its limit system in the sense that V (x) → V∞ ∈ R+ as |x| → +∞. Subsequently, Hu-Wu-Tang [26] established the existence of least energy sign-changing solutions of (1.3). For more recent results, we refer to [14, 34, 36, 37, 39, 47, 48]. Note that the papers mentioned above on system (1.2) assume ω ∈ R as a fixed parameter to study nontrivial solutions. Alternatively, we can search for solutions with the prescribed L2-norm for system (1.1). This approach seems to be meaningful from the physical point of view because of the conservation of mass. In the present study, we focus on finding normalized solutions to (1.1), i.e. a couple (u, λ) ∈ H1(R3)× R satisfies (1.1) together with the normalization condition∫ R3 |u|2 dx = c (1.4) for a priori given c > 0. As we know, for each u ∈ H1(R3), there exists a unique solution φ = φu to the second equation of (1.1) satisfying φu(x) = 1 4π ∫ R3 1− e−|x−y| |x− y| u2(y) dy. Then, system (1.1) is reduced into an equivalent integro-differential form −∆u+ φuu = λu+ µ|u|p−2u+ |u|4u in R3. (1.5) It is standard that for any c > 0, a solution of (1.5)-(1.4) can be regarded as a critical point of the corresponding Energy functional I(u) := 1 2 ∫ R3 |∇u|2 dx+ 1 4 ∫ R3 φuu 2 dx− µ p ∫ R3 |u|p dx− 1 6 ∫ R3 |u|6 dx, restricted to S(c) := { u ∈ H1(R3) : ∫ R3 u2 dx = c } . EJDE-2023/56 CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEM 3 Then the parameter λ ∈ R appears as a Lagrange multiplier. It is easy to verify that I(u) is a well-defined and C1 functional on H1(R3). Recently, there are numerous contributions flourishing within this topic, for instance, see [2, 3, 4, 5, 10, 18, 20, 27, 28, 29, 42, 43, 45]. Definition 1.1. We say that ũ ∈ S(c) is a ground state solution of (1.5) if it is a solution having minimal energy among all the solutions which belong to S(c), i.e., dI|S(c)(ũ) = 0, I(ũ) = inf{I(u) : dI|S(c)(u) = 0, u ∈ S(c)}. Afonso-Siciliano [1] considered the existence of normalized solutions for the Schrödinger-Bopp-Podolsky system in bounded domains under Neumann bound- ary conditions. He-Li-Chen [25] investigated the following system, −∆u+ ωu+ φu = |u|p−2u in R3, −∆φ+ a2∆2φ = 4πu2 in R3, ‖u‖2L2 = ρ, (1.6) where ω ∈ R, a > 0, and ρ > 0. By the minimizing method, they obtained the existence of normalized solutions for ω > 0, a = 1, and p ∈ (2, 10 3 ), in which the corresponding functional is bounded from below on S(c). Ramos-Siciliano [40] proved that if 2 < p < 3, ρ > 0 is sufficiently small or if 3 < p < 10 3 , ρ > 0 is sufficiently large, then system (1.6) admits a least energy solution. Moreover, in the case of 2 < p < 14 5 and ρ > 0 small enough, the least energy solutions are radially symmetric up to translation and converge to a least energy solution of the Schrödinger-Poisson-Slater system under the same L2-norm constraint. We remark that the above papers do not involve the L2-supcritical case where p > 10 3 . Since in such a situation, the classical methods for dealing with L2- supercritical problems fail due to the fact that φu is not homogeneous, which is difficult for us to make use of the scaling of type t 7→ tαu(tβ ·) for α, β ∈ R and t > 0. Moreover, the appearance of the term ∫ R3 ∫ R3 e −|x−y|u2(x)u2(y) dx dy in the corresponding Pohozaev identity is another obstacle to deal with. It is worthy to note that in the case of a = 0, this problem reduces to the well-known Schrödinger- Poisson-Slater system −∆u+ b1(|x|−1 ∗ |u|2)u+ λu = b2|u|p−2u in R3, ‖u‖2L2 = c, (1.7) where b1, b2 ∈ R, and p ∈ ( 10 3 , 6]. Bellazzini-Jeanjean-Luo [6] proved that if b1, b2 > 0, then (1.7) admits a solution of mountain pass type for c > 0 sufficiently small by using a mountain-pass argument. Recently, this result has been developed by Jeanjean-Le [29] under different assumptions on b1, b2, and p. Li-Zhang [32] investigated system (1.1) with a Sobolev critical term. For p ∈ ( 10 3 , 6), by applying a mountain-pass argument, they established the existence of positive normalized ground state solutions to (1.1) for large µ > 0 and small c > 0. For p ∈ (2, 10 3 ], by combining the mountain pass theorem with Lebesgue dominated convergence theorem, they proved the existence of normalized ground state solutions for large µ > 0 and small c > 0. Because of the critical term |u|4u, it is not difficult to check that I|S(c) is un- bounded from below. However, as we see, the combined action of L2 subcritical term µ|u|p−2u and the nonlocal term φuu creates a geometry of local minima of 4 Y. X. LI, X. J. CHANG, Z. FENG EJDE-2023/56 I on S(c) for c > 0 small enough. Based on [28, 43], there is a natural question whether I|S(c) has a critical point which is a local minimizer in the case where p ∈ (2, 8 3 ). This constitutes the main motivation of this study and our goal is to make an effort to find a positive answer to this question. For u ∈ S(c), we set ut(x) := t3/2u(tx), t > 0, x ∈ R3. A direct calculation leads to Φu(t) := I(ut) = t2 2 ∫ R3 |∇u|2 dx+ t 16π ∫ R3 ∫ R3 1− e− 1 t |x−y| |x− y| u2(x)u2(y) dx dy − µ t 3(p−2) 2 p ∫ R3 |u|p dx− t6 6 ∫ R3 |u|6 dx, (1.8) which is the so-called fiber map and plays an important role in the discussion of the geometrical structure of the functional I. At this stage, we introduce the related Pohozaev manifold defined by Λ(c) := {u ∈ S(c) : Q(u) = 0}, where Q(u) := d dt ∣∣ t=1 I(ut) = ∫ R3 |∇u|2 dx+ 1 16π ∫ R3 ∫ R3 1− e−|x−y| |x− y| u2(x)u2(y) dx dy − 1 16π ∫ R3 ∫ R3 e−|x−y|u2(x)u2(y) dx dy − 3µ (p− 2) 2p ∫ R3 |u|p dx − ∫ R3 |u|6 dx. (1.9) As mentioned earlier, for any fixed µ > 0, we can find a c0 = c0(µ) > 0 such that, for any c ∈ (0, c0) there exists an open set V (c) ⊂ S(c) with the property m(c) := inf u∈V (c) I(u) < 0 < inf u∈∂V (c) I(u), where V (c) := {u ∈ S(c) : ‖∇u‖22 < ρ0}, ∂V (c) := {u ∈ S(c) : ‖∇u‖22 = ρ0} for a suitable ρ0 > 0 only depending on c0 > 0. In the process of minimization, the key difficulty is the lack of compactness of the bounded minimizing sequence {un} ⊂ V (c) and the most critical step is to prove the strong subadditivity inequality m(c) < m(c1) +m(c2) for all 0 < c1, c2 < c, (1.10) which is a sufficient condition to ensure that any minimizing sequence on V (c) is relatively compact. Moreover, (1.10) is a stronger version of the so-called weak subadditivity inequality m(c) ≤ m(c1) +m(c2) for all 0 < c1, c2 < c. (1.11) EJDE-2023/56 CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEM 5 However, because of p ∈ (2, 8 3 ) and the existence of the nonlocal term ∫ R3 φuu 2 dx, the method introduced in [28, Lemma 2.6] becomes invalid. In fact, if we do the scaling v = √ θu, it is impossible to obtain that m(θα) ≤ θm(α), θ > 1, α > 0. Following [7], we introduce the condition the function c→ m(c) c is strictly decreasing. (1.12) From this assumption it follows c1 c m(c) < m(c1), c− c1 c < m(c− c1). That is, m(c) = c1 c m(c) + c− c1 c m(c) < m(c1) +m(c− c1), whenever 0 < c1 < c. However, verifying condition (1.12) is not easy since the function c→ m(c) c has oscillatory behavior, even in a neighborhood of the origin. To overcome this difficulty, we adapt the techniques developed by Bellazzini-Siciliano [7, 8] as to finding sufficient conditions to avoid dichotomy. As we see, the presence of the term ∫ R3 φuu 2 dx has a significant impact on the geometry of Φu(t) and on the existence of ground state solutions. The existence of a normalized ground state solution for the nonlinear Schrödinger equation with a Sobolev critical term and a L2-subcritical perturbation was discussed in [28], where the local minima of the constraint functional is exactly a ground state. However, it is not trivial for our case due to the complex structure of the fiber map. For this situation, we turn our attention to the Pohozaev manifold Λ(c) which contains the solutions of (1.1), see Lemma 2.3, to search for a ground state by taking the minimization in the set of solutions. Our main result reads as follows. Theorem 1.2. Let p ∈ (2, 8/3). For any µ > 0 there exists a c0 = c0(µ) > 0 such that, for any c ∈ (0, c0), I(u) restricted to S(c) has a critical point uc at a negative level m(c) < 0 which is an interior local minimizer of I(u) in the set V (c). Moreover, system (1.1) admits a ground state solution on S(c). Before concluding this section, we would like to summarize new features in this study. • The approach which we use for Theorem 1.2 distinguishes from those de- scribed in the literature, for example, see [32, Theorem 1.2]. In fact, our arguments are based on the minimizing method instead of the mountain pass theorem. Moreover, our arguments work for all µ > 0 and we do not need to assume the range of the Lagrange multiplier λ in the first step. • To show that the minimizing sequences for m(c) are relatively compact, we make use of the strong subadditivity inequality (1.10). We can not just take the same steps as shown in [28] because of the presence of the term∫ R3 φuu dx. Therefore, we turn to develop another method to ensure the inequality (1.10) to be true. Namely, we take into account how to guarantee the condition (1.12). • The exponential term in φu makes the fiber map Φu(t) more complicated since it exists in the first and second derivatives. Thus we cannot follow [28] directly to draw a conclusion that any ground state is contained in 6 Y. X. LI, X. J. CHANG, Z. FENG EJDE-2023/56 V (c). To show the existence of the ground state solution, we take a series of solutions in Λ(c) and obtain the bounded Palais-Smale sequences which are the minimizers of I(u) on S(c). This article is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used later. In Section 3, we prove our main result by clari- fying the local minima structure and showing the convergence, up to a translation, of all minimizing sequences for the functional I(u) on V (c). 2. Preliminary results Throughout this paper, for any 1 ≤ s < ∞, we denote by Ls(R3) the usual Lebesgue space with norm ‖u‖ss := ∫ R3 |u|s dx. We use C∞0 (R3) to denote the space of the functions infinitely differentiable with compact support in R3. We denote by C1, C2, . . . the positive constants whose values possibly vary from line to line. The open ball in R3 is denoted by B(x,R) with the center at x and the radius R. We start with the Hardy-Littlewood-Sobolev inequality [33]:∣∣ ∫ RN ∫ RN f(x)g(y) |x− y|λ dx dy ∣∣ ≤ C(N,λ, p, q)‖f‖p‖g‖q, (2.1) where f ∈ Lp(RN ), g ∈ Lq(RN ), p, q > 1, 0 < λ < N , and 1 p + 1 q + λ N = 2. The following Gagliardo-Nirenberg inequality can be found in [46]: ‖u‖p ≤ K1/p GN‖∇u‖ τ 2‖u‖1−τ2 , (2.2) where N ≥ 3, p ∈ [2, 2N N−2 ], and τ = N( 1 2 − 1 p ). We recall the optimal Sobolev constant S > 0, see [15], which is S = inf u∈D1,2(R3), u 6=0 ‖∇u‖22 ‖u‖26 , where D1,2(R3) := {u ∈ L6(R3) : |∇u| ∈ L2(R3)} is the completion of C∞0 (R3) with the norm ‖u‖D1,2(R3) = ‖∇u‖2. The Hilbert space defined by D := {u ∈ D1,2(R3) : ∆u ∈ L2(R3)} is the completion of C∞0 (R3) with respect to the norm ‖u‖2D = ‖∆u‖22 + ‖∇u‖22. It is easy to show that D is continuously embedded into D1,2(R3), see [22]. Lemma 2.1. [22, Lemma 3.4] For every u ∈ H1(R3) we have (i) for every y ∈ R3, φu(·+y) = φu(·+ y); (ii) φu ≥ 0; (iii) φu ∈ D; (iv) ‖φu‖6 ≤ C‖u‖2H1(R3); and (v) if vn ⇀ v in H1(R3), then φvn ⇀ φv in D. Lemma 2.2. Let u ∈ S(c). Then we have EJDE-2023/56 CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEM 7 (i) There exists a constant KH > 0 such that∫ R3 ∫ R3 |u(x)|2|u(y)|2 |x− y| dx dy ≤ KH‖∇u‖2 c3/2. (ii) There exists a constant KGN > 0 such that ‖u‖pp ≤ KGN‖∇u‖ 3(p−2) 2 2 c 6−p 4 . Proof. From (2.1) and (2.2) it follows that∫ R3 ∫ R3 |u(x)|2|u(y)|2 |x− y| dx dy ≤ K1‖u‖412/5 ≤ KH‖∇u‖2‖u‖32, which implies (i). In view of (2.2), we derive ‖u‖pp ≤ KGN‖∇u‖ 3(p−2) 2 2 ‖u‖ 6−p 2 2 , which leads to (ii). � Lemma 2.3. Let µ > 0 and 2 < p < 6. If (u, λ) ∈ H1(R3)× R weakly solves −∆u+ φuu = λu+ µ|u|p−2u+ |u|4u, (2.3) then Q(u) = 0, where Q(u) is defined by (1.9). Proof. Using the Pohozaev identity in [19, Lemma 4.2] yields 1 2 ∫ R3 |∇u|2 dx+ 5 4 ∫ R3 φuu 2 dx+ 1 16π ∫ R3 ∫ R3 e−|x−y|u2(x)u2(y) dx dy = 3λ 2 ∫ R3 |u|2 dx+ 3µ p ∫ R3 |u|p dx+ 1 2 ∫ R3 |u|6 dx = 0. (2.4) Multiplying both sides of (2.3) by u and integrating on R3 leads to∫ R3 |∇u|2 dx+ ∫ R3 φuu 2 dx = λ ∫ R3 |u|2 dx+ µ ∫ R3 |u|p dx+ ∫ R3 |u|6 dx. (2.5) By combining (2.4) and (2.5), we obtain∫ R3 |∇u|2 dx+ 1 16π ∫ R3 ∫ R3 1− e−|x−y| |x− y| u2(x)u2(y) dx dy = 1 16π ∫ R3 ∫ R3 e−|x−y|u2(x)u2(y) dx dy + 3µ (p− 2) 2p ∫ R3 |u|p dx+ ∫ R3 |u|6 dx. Therefore, we arrive at the desired result. � To ensure the condition (1.12), we define A(u) := ∫ R3 |∇u|2 dx, B(u) := ∫ R3 φuu 2 dx, C(u) := ∫ R3 |u|p dx, D(u) := ∫ R3 |u|6 dx, T (u) := 1 4 B(u)− µ p C(u)− 1 6 D(u). Thus, the functional I(u) can be simply re-written as I(u) = 1 2 A(u) + T (u). Next we recall two definitions introduced in [7]. 8 Y. X. LI, X. J. CHANG, Z. FENG EJDE-2023/56 Definition 2.4. Let u ∈ H1(R3) with u 6= 0. A continuous path gu : θ ∈ R+ 7→ gu(θ) ∈ H1(R3) such that gu(1) = u is said to be a scaling path of u if Θgu(θ) := ‖gu(θ)‖22/‖u‖22 is differentiable and Θ′gu(1) 6= 0. We denote by Gu the set of the scaling paths of u. Definition 2.5. Let u 6= 0 be fixed and gu ∈ Gu. We say that the scaling path gu is admissible for the functional I if hgu(θ) := I(gu(θ))−Θgu(θ)I(u), θ ≥ 0 is a differentiable function. The following lemma is regarding the splitting properties of the term T , see [25, Lemma 2.8] and [22, Lemma B.2]. Lemma 2.6. If p ∈ (2, 10/3), we let {un} ⊂ V (c) be a minimizing sequence for m(c) such that un ⇀ u 6= 0. Then T satisfies the following properties: (i) T (un − u) + T (u) = T (un) + on(1); and (ii) T (αn(un − u))− T (un − u) = on(1), where αn = ‖un‖22 − ‖u‖22/‖un − u‖22. The following proposition provides us a useful criterion for the condition (1.12), [7, Theorem 2.1]. Proposition 2.7. Let T ∈ C1(H1(R3),R) satisfy Lemma 2.6 (i) and (ii). Assume that for every c > 0, all the minimizing sequences {un} for m(c) have a weak limit up to translations different from zero. Assume that (1.11) and the following conditions hold −∞ < m(c) < 0, for all c > 0 (I(0) = 0), (2.6) c 7→ m(c) is continuous, (2.7) lim c→0 m(c) c = 0. (2.8) Then, for each c > 0, the set M(c) = ∪c̃∈(0,c]{u ∈ S(c̃) : I(u) = m(c̃)} is nonempty. In addition, if ∀u ∈M(c), ∃gu ∈ Gu is admissible such that d dθ hgu(θ)|θ=1 6= 0, (2.9) then the condition (1.12) holds. Moreover, if {un} is a minimizing sequence weakly convergent to a certain u (necessarily 6= 0), then ‖un − u‖H1(R3) → 0 and I(u) = m(c). 3. Proof of Theorem 1.2 Throughout the whole section, we assume that 2 < p < 8 3 , from which we have 0 < 3(p−2) 2 < 1. Note that I(u) ≥ 1 2 ‖∇u‖22 − µKGN p c 6−p 4 ‖∇u‖ 3(p−2) 2 2 − 1 6S3 ‖∇u‖62 (3.1) for any u ∈ S(c). Now we consider the function h : R+ → R, defined by hc(t) := 1 2 t2 − µKGN p c 6−p 4 t 3(p−2) 2 − 1 6S3 t6 EJDE-2023/56 CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEM 9 = t2[ 1 2 − µKGN p c 6−p 4 t 3(p−2) 2 −2 − 1 6S3 t4]. Since µ > 0 and 3(p−2) 2 < 1, we have hc(0 +) = 0− and hc(+∞) = −∞. Moreover, the following properties hold for hc. Lemma 3.1 ([28, Lemma 2.1]). For any µ > 0 there exist a c0 = c0(µ) > 0 and ρ0 := ρc0 > 0 such that hc0( √ ρ0) = 0 and hc( √ ρ0) > 0 hold for any c ∈ (0, c0), where c0 and ρ0 are explicitly given by c0 := ( 1 2K )3/2 > 0, with K := µ p KGN [ − 3(3p− 10)µKGNS3 4p ] 3p−10 3(6−p) + 1 6S3 [ − 3(3p− 10)µKGNS3 4p ] 8 3(6−p) > 0, ρ0 := [ − 3(3p− 10)µKGNS3 4p ] 4 3(6−p) c 1/3 0 . Lemma 3.2 ([28, Lemma 2.2]). Let (c1, ρ1) ∈ (0,∞)×(0,∞) satisfy hc1( √ ρ1) ≥ 0. Then for any c2 ∈ (0, c1] there holds hc2( √ ρ2) ≥ 0, if ρ2 ∈ [c2 c1 ρ1, ρ1 ] . Remark 3.3. For p ∈ (2, 10/3), from the expression of hc(t) we can deduce that hc(0 +) = 0− and hc(+∞) = −∞, which means that Lemma 3.1 also holds in such a case. However, taking into account the geometrical structure of the fiber map Φu(t), we have to reduce the range of p to p ∈ (2, 8 3 ), see Lemma 3.6 below. Remark 3.4. According to Lemmas 3.1 and 3.2, it is not difficult to see that hc0( √ ρ0) = 0 and hc( √ ρ0) > 0 for all c ∈ (0, c0). Lemma 3.5. For c ∈ (0, c0), I(u) restricted to Λ(c) is coercive on H1(R3). Namely, if {un} ⊂ H1(R3) satisfies ‖un‖H1(R3) → +∞, then I(un)→ +∞. Proof. Let u ∈ Λ(c). Taking into account Q(u) = 0, we have A(u) + 1 4 B(u)− 1 16π ∫ R3 ∫ R3 e−|x−y|u2(x)u2(y) dx dy = 3µ(p− 2) 2p C(u) +D(u). (3.2) From Lemma 2.2 (i), there exists a constant C1 > 0 such that B(u) ≤ C1‖∇u‖2c3/2. In view of (3.2), there exists a constant C2 > 0 such that D(u) ≤ A(u) + C2‖∇u‖2c3/2. This together with Lemma 2.2 (ii), for some C3 > 0, leads to I(u) = 1 2 A(u) + 1 4 B(u)− µ p C(u)− 1 6 D(u) ≥ 1 2 A(u)− µ p C(u)− 1 6 A(u)− C2 6 A(u) 1 2 c3/2 10 Y. X. LI, X. J. CHANG, Z. FENG EJDE-2023/56 ≥ 1 3 A(u)− C3A(u) 3(p−2) 4 c 6−p 4 − C2 6 A(u) 1 2 c3/2, from which we e complete the proof. � Define Bρ0 := {u ∈ H1(R3) : ‖∇u‖22 < ρ0}, V (c) := S(c) ∩Bρ0 and consider a minimization problem: m(c) = inf u∈V (c) I(u), for any c ∈ (0, c0). Lemma 3.6. Let c ∈ (0, c0). Then the following three assertions hold. (i) m(c) = infu∈V (c) I(u) < 0 < infu∈∂V (c) I(u). (ii) The function c 7→ m(c) is a continuous mapping. (iii) For all α ∈ (0, c), we have m(c) ≤ m(α) +m(c− α). Proof. (i) For any u ∈ ∂V (c) we have ‖∇u‖22 = ρ0. From (3.1) it follows that I(u) ≥ hc(‖∇u‖2) = hc( √ ρ0) > 0. Taking into account 3(p−2) 2 < 1, we have Φu(t) → 0− as t → 0. Therefore, there exists t0 < 1 small enough such that ‖∇ut0‖22 = t20‖∇u‖22 < ρ0 and I(ut0) = Φu(t0) < 0, which means m(c) < 0. (ii) For any c ∈ (0, c0), let {cn} ⊂ (0, c0) satisfy cn → c as n → ∞. Recall the definition of m(cn) < 0. For any ε > 0 small enough, there exists {un} ⊂ V (cn) such that I(un) ≤ m(cn) + ε, I(un) < 0. Let vn := ( c cn )√1/2 un. Then vn ∈ S(c) and by similar arguments as described in [28, Lemma 2.6], we see that vn ∈ V (c). Furthermore, we have m(c) ≤ I(vn) = 1 2 c cn A(un) + 1 4 ( c cn )2B(un)− µ p ( c cn ) p 2C(un)− 1 6 ( c cn )3D(un) = I(un) + on(1) ≤ m(cn) + ε+ on(1). Similarly, for any ε > 0 there exists u ∈ V (c) such that I(u) ≤ m(c) + ε, I(u) < 0. Let vn := ( cn c )√1/2 u. Then vn ∈ V (cn). Processing in an analogous manner, we can obtain m(cn) ≤ I(un) = [I(un)− I(u)] + I(u) ≤ m(c) + ε+ on(1). In view of ε > 0 being arbitrary, we have m(cn) → m(c) as n → ∞ which implies (ii). (iii) By the fact that C∞0 (R3) is dense in H1(R3), for any ε > 0 there exist u1 ∈ C∞0 (R3) ∩ V (α) and u2 ∈ C∞0 (R3) ∩ V (c− α) satisfying I(u1) ≤ m(α) + ε 2 , I(u2) ≤ m(c− α) + ε 2 , (3.3) I(u1) < 0, I(u2) < 0. (3.4) EJDE-2023/56 CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEM 11 Moreover, for any n ∈ N, by a translation, we may assume that dist(suppu1, suppu2) > n. By Lemma 3.2, we have hα( √ ρ) ≥ 0 for any ρ ∈ [αc ρ0, ρ0] and hc−α( √ ρ) ≥ 0 for each ρ ∈ [ c−αc ρ0, ρ0]. Hence, we can deduce from (3.4) that ‖∇u1‖22 < α c ρ0, ‖∇u2‖22 < c− α c ρ0. Let u = u1 + u2. It is easy to verify that ‖u‖22 = ‖u1‖22 + ‖u2‖22 and A(u) = A(u1) +A(u2). Thus we have ‖u‖22 = c, ‖∇u‖22 < ρ0, which means u ∈ V (c). Notice that |u(x)|2|u(y)|2 = |u1(x) + u2(x)|2|u1(y) + u2(y)|2 = |u1(x)|2|u1(y)|2 + |u1(x)|2|u2(y)|2 + |u2(x)|2|u1(y)|2 + |u2(x)|2|u2(y)|2 + 2|u1(x)|2u1(y)u2(y) + 2|u2(x)|2u1(y)u2(y) + 2|u1(y)|2u1(x)u2(x) + 2|u2(y)|2u1(x)u2(x) + 4u1(x)u2(x)u1(y)u2(y). Then we can deduce that∫ R3 ∫ R3 1− e−|x−y| |x− y| |u1(x)|2|u2(y)|2 dx dy = ∫ R3 ∫ R3 1− e−|x−y| |x− y| |u1(y)|2|u2(x)|2 dx dy ≤ ∫ R3 ∫ R3 |u1(y)|2|u2(x)|2 |x− y| dx dy = ∫ suppu1 ∫ suppu2 |u1(y)|2|u2(x)|2 |x− y| dx dy ≤ α(c− α) n , ∫ R3 ∫ R3 1− e−|x−y| |x− y| |u1(x)|2|u1(y)||u2(y)| dx dy = ∫ R3 ∫ R3 1− e−|x−y| |x− y| |u1(y)|2|u1(x)||u2(x)| dx dy ≤ αc 2n , ∫ R3 ∫ R3 1− e−|x−y| |x− y| |u2(x)|2|u1(y)||u2(y)| dx dy = ∫ R3 ∫ R3 1− e−|x−y| |x− y| |u2(y)|2|u1(x)||u2(x)| dx dy ≤ (c− α)c 2n , and ∫ R3 ∫ R3 1− e−|x−y| |x− y| |u1(x)||u2(x)||u1(y)||u2(y)| dx dy ≤ ∫ R3 ∫ R3 |u1(x)||u2(x)||u1(y)||u2(y)| |x− y| dx dy ≤ c2 4n . 12 Y. X. LI, X. J. CHANG, Z. FENG EJDE-2023/56 Therefore, B(u) = B(u1) +B(u2) + on(1). Clearly, it holds C(u) = C(u1) + C(u2), D(u) = D(u1) +D(u2). Therefore, from (3.3) it follows that m(c) ≤ I(u) = I(u1) + I(u2) + on(1) ≤ m(α) +m(c− α) + ε+ on(1). Since ε > 0 is arbitrary, we have m(c) ≤ m(α) +m(c− α). Consequently, we have arrived at (iii). � Lemma 3.7. Let {vn} ⊂ Bρ0 satisfy ‖vn‖p → 0. Then there exists a β0 > 0 such that I(vn) ≥ β0‖∇vn‖22 + on(1). Proof. By a direct calculation, we have I(vn) ≥ 1 2 ‖∇vn‖22 − 1 6 ‖vn‖66 + on(1) ≥ 1 2 ‖∇vn‖22 − 1 6S3 ‖∇vn‖62 + on(1) ≥ ‖∇vn‖22[ 1 2 − 1 6S3 ρ2 0] + on(1). Since hc0( √ ρ0) = 0, we obtain β0 := 1 2 − 1 6S3 ρ2 0 = µKGN p c 6−p 4 0 ρ 3(p−2) 4 −1 0 > 0. � Lemma 3.8. For any c ∈ (0, c0), let {un} ⊂ V (c) be a minimizing sequence for m(c) such that un ⇀ u in H1(R3) as n→∞. Then u 6= 0. Proof. To show that there exist a β1 > 0 and a sequence {yn} ⊂ R3 such that for some R > 0 it holds ∫ B(yn,R) |un|2 dx ≥ β1 > 0, (3.5) we argue by contradiction that (3.5) does not hold. According to [35, Lemma I.1], for 2 < p < 6 we have ‖un‖Lp(R3) → 0 as n→∞. Then, from Lemma 3.7 it follows that I(un) ≥ on(1). This contradicts the fact that m(c) < 0. From (3.5), we know that there exist some C > 0 and a sequence {yn} ⊂ R3 such that ∫ B(0,R) |un(· − yn)|2 dx ≥ C > 0. Because of the Rellich compactness theorem, we have un(x− yn) ⇀ u 6= 0 in H1(R3), which enables us to arrive at the desired result. � EJDE-2023/56 CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEM 13 Next we are going to verify all conditions described in Proposition 2.7 for pre- senting the proof of Theorem 1.2. In view of Lemmas 2.6, 3.6, and 3.8, it suffices to prove (2.8) and (2.9). Lemma 3.9. Assume that c ∈ (0, c0). Then the function c 7→ m(c) satisfies (2.8). Proof. Since m(c) < 0, we have m̃(c) c ≤ m(c) c < 0, where m̃(c) := inf u∈V (c) Ĩ(u), Ĩ(u) := 1 2 ∫ R3 |∇u|2 dx− µ p ∫ R3 |u|p dx− 1 6 ∫ R3 |u|6 dx. Thus, it suffices to show that m̃(c) c → 0 as c → 0. Indeed, Ĩ(u) is the functional associated with the following Schrödinger equation with combined nonlinearities −∆u = λu+ µ|u|p−2u+ |u|4u in R3, with the normalized condition ‖u‖22 = c. According to [28, Theorem 1.2], for any c ∈ (0, c0) there exists uc ∈ V (c) such that m̃(c) = Ĩ(uc) < 0. Moreover, we know that the sequence {uc}c∈(0,c0) is bounded in D1,2(R3) as c→ 0 and uc satisfies the following equation in the weak sense −∆uc = λcuc + µ|uc|p−2uc + |uc|4uc in R3, (3.6) from which we deduce that λc 2 = ∫ R3 |∇uc|2 dx− µ ∫ R3 |uc|p dx− ∫ R3 |uc|6 dx 2 ∫ R3 |uc|2 dx ≤ 1 2 ∫ R3 |∇uc|2 dx− µ p ∫ R3 |uc|p dx− 1 6 ∫ R3 |uc|6 dx∫ R3 |uc|2 dx = Ĩ(uc) c = m̃(c) c < 0. To show that limc→0 λc = 0, we argue by contradiction: assume that there exists a sequence cn → 0 such that λcn < −C for some C ∈ (0, 1). Since the minimizers un := ucn satisfies (3.6), there exist some C1, C2 > 0 such that C‖un‖2H1(R3) ≤ ∫ R3 |∇un|2 dx+ C ∫ R3 |un|2 dx < ∫ R3 |∇un|2 dx− λcn ∫ R3 |un|2 dx = µ ∫ R3 |un|p dx+ ∫ R3 |un|6 dx ≤ C1‖un‖pH1(R3) + C2‖un‖6H1(R3). This indicates that there exists C3 > 0 such that ‖∇un‖22 > C3 > 0 because p > 2. Hence, from Remark 3.4 it follows that 0 > Ĩ(un) ≥ 1 2 ‖∇un‖22 − µKGN p c 6−p 4 n ‖∇un‖ 3(p−2) 2 2 − 1 6S3 ‖∇un‖62 14 Y. X. LI, X. J. CHANG, Z. FENG EJDE-2023/56 = ‖∇un‖22 (1 2 − µKGN p c 6−p 4 n ‖∇un‖ 3(p−2) 2 −2 2 − 1 6S3 ‖∇un‖42 ) ≥ C3 (1 2 − µKGN p c 6−p 4 n ρ 3(p−2) 4 −1 0 − 1 6S3 ρ2 0 ) > 0, which yields a contradiction. � Lemma 3.10. Let c ∈ (0, c0). Then the following strict subadditivity inequality holds, m(c) < m(c1) +m(c2), where c = c1 + c2 and 0 < c1, c2 < c. Proof. Note that from Proposition 2.7, we just need to show the condition (2.9) holds. For u ∈M(c), without loss of generality, we suppose that there exists some c̃ ∈ (0, c] such that ‖u‖22 = c̃ and I(u) = m(c̃). Since u is a minimizer of I(u) on V (c̃), we deduce from Lemma 2.3 that A(u) + 1 4 B(u)− 1 16π ∫ R3 ∫ R3 e−|x−y|u2(x)u2(y) dx dy− 3µ(p− 2) 2p C(u)−D(u) = 0. (3.7) For u 6= 0 we compute hgu(θ) by considering the family of scaling paths of u parameterized with β ∈ R given by uθ(x) := θ 1+3β 2 u(θβx). By a straightforward computation, we have A(uθ) = θ1+2βA(u), B(uθ) = θ2+βH(u)− θ2+β 4π ∫ R3 ∫ R3 e− |x−y| θβ |x− y| u2(x)u2(y) dx dy, C(uθ) = θ (1+3β)p 2 −3βC(u), D(uθ) = θ3(1+2β)D(u), ‖uθ‖22 = θ‖u‖22, where H(u) = 1 4π ∫ R3 ∫ R3 u2(x)u2(y) |x−y| dx dy. Let hgu(θ) = f(θ, u) := I(uθ)− θI(u). Then hgu(θ) = f(θ, u) = 1 2 (θ1+2β − θ)A(u) + 1 4 [ θ2+βH(u) − θ2+β 4π ∫ R3 ∫ R3 e− |x−y| θβ |x− y| u2(x)u2(y) dx dy − θB(u) ] − µ p (θ (1+3β)p 2 −3β − θ)C(u)− 1 6 (θ3(1+2β) − θ)D(u). (3.8) Moreover, we can deduce that f ′θ(θ, u) = 1 2 ( (1 + 2β)θ2β − 1 ) A(u) + 1 4 [ (2 + β)θ1+βH(u) − (2 + β)θ1+β 4π ∫ R3 ∫ R3 e− |x−y| θβ |x− y| u2(x)u2(y) dx dy − θ2+β 4π β θβ+1 ∫ R3 ∫ R3 e− |x−y| θβ u2(x)u2(y) dx dy −B(u) ] − µ p ( ( (1 + 3β)p 2 − 3β)θ (1+3β)p 2 −3β−1 − 1 ) C(u) − 1 6 ( (3(1 + 2β))θ3(1+2β)−1 − 1 ) D(u), EJDE-2023/56 CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEM 15 which leads to f ′θ(1, u) = βA(u) + 1 4 [ − β 4π ∫ R3 ∫ R3 e−|x−y|u2(x)u2(y) dx dy + (1 + β)B(u) ] − µ p ( (1 + 3β)p 2 − 3β − 1)C(u)− 1 6 (3(1 + 2β)− 1)D(u). (3.9) Now it remains to show that the admissible scaling path satisfies h′gu(1) 6= 0. Again, we process by way of contradiction: assume that there exists a sequence {un} ⊂ M(c) with ‖un‖22 = cn ≤ c and cn → 0 such that for all β ∈ R, we have f ′θ(1, un) = 0. That is, βA(un) + 1 4 [ − β 4π ∫ R3 ∫ R3 e−|x−y|u2 n(x)u2 n(y) dx dy + (1 + β)B(un) ] − µ p ( (1 + 3β)p 2 − 3β − 1)C(un)− 1 6 (3(1 + 2β)− 1)D(un) = 0. (3.10) Combining (3.7) and (3.10) yields 1 4 B(un) = µ(p− 2) 2p C(un) + 1 3 D(un), (3.11) B(un) = 2A(un)− 1 8π ∫ R3 ∫ R3 e−|x−y|u2 n(x)u2 n(y) dx dy. (3.12) Moreover, from the continuity of m(c) and the Gagliardo-Nirenberg inequality (2.2), we have I(un) = m(cn)→ 0, A(un), B(un), C(un), D(un)→ 0. (3.13) We need to consider three cases. Case 1: 2 < p < 12/5. From the Hardy-Littlewood-Sobolev inequality, the inter- polation inequality, the Sobolev embedding theorem and (3.11), it follows that B(un) = 1 4π ∫ R3 ∫ R3 1− e−|x−y| |x− y| u2 n(x)u2 n(y) dx dy ≤ 1 4π ∫ R3 ∫ R3 u2 n(x)u2 n(y) |x− y| dx dy ≤ C‖un‖412/5 ≤ C‖un‖ 6p 6−p p ‖un‖ 12−5p 3p 6 = CC(un) 6 6−pA(un) 12−5p 6p ≤ C1B(un) 6 6−pA(un) 12−5p 6p . This leads to 1 ≤ C1B(un) p 6−pA(un) 12−5p 6p , which yields a contradiction with (3.13). Case 2: p = 12 5 . Due to (3.11), we obtain ‖un‖12/5 12/5 ≤ 3 µ B(un) ≤ C2‖un‖412/5, which is impossible because of the fact ‖un‖12/5 → 0. 16 Y. X. LI, X. J. CHANG, Z. FENG EJDE-2023/56 Case 3: 12/5 < p < 8/3. From (3.11) it follows that ‖un‖pp ≤ p 2µ(p− 2) B(un) ≤ C3‖un‖412/5 ≤ C3‖un‖ 2(5p−12) 3(p−2) 2 ‖un‖ 2p 3(p−2) p . This leads to 1 ≤ C3c 5p−12 3(p−2) n ‖un‖ p(8−3p) 3(p−2) p , which yields another contradiction with (3.13). All the hypotheses of Proposition 2.7 have been verified and thus we have arrived at the desired result. � Proof of Theorem 1.2. For any c ∈ (0, c0), we assume that {un} ⊂ V (c) satisfies I(un)→ m(c). By Lemma 3.8, there exists a sequence {yn} ⊂ R3 such that un(x− yn) ⇀ u 6= 0 in H1(R3). We start by showing that wn(x) := un(x− yn)− u(x)→ 0 in H1(R3). Clearly, we can see that ‖un‖22 = ‖wn‖22 + ‖u‖22 + on(1), ‖∇un‖22 = ‖∇wn‖22 + ‖∇u‖22 + on(1), I(un) = I(wn) + I(u) + on(1). The last equality holds because of the translational invariance. Then we claim that ‖wn‖22 → 0. Let ‖u‖22 = c1 > 0. It suffices to show that c1 = c. We assume by contradiction that c1 < c. Since we have, for n large enough, ‖wn‖22 ≤ c and ‖∇wn‖22 ≤ ‖∇un‖22 < ρ0. Then wn ∈ V (‖wn‖22) and I(wn) ≥ m(‖wn‖22), which implies that m(c) = I(wn) + I(u) + on(1) ≥ m(‖wn‖22) + I(u) + on(1). By Lemma 3.6 (ii), we have m(c) ≥ m(c− c1) + I(u). Moreover, we see that u ∈ V (c1). Then I(u) ≥ m(c1), and from from Lemma 3.10 it follows that m(c) ≥ m(c− c1) +m(c1) > m(c), which is impossible. Hence the claim follows and ‖u‖22 = c. Now we show that ‖∇wn‖22 → 0. Since u 6= 0, we have ‖∇wn‖22 ≤ ‖∇un‖22 < ρ0 for n large enough. Thus {wn} ⊂ Bρ0 and {wn} is bounded in H1(R3). By Lemma 2.2 (ii), recalling ‖wn‖22 → 0, we obtain ‖wn‖pp → 0. Then, from Lemma 3.7 we can deduce that I(wn) ≥ β0‖∇wn‖22 + on(1) for β0 > 0. (3.14) Since u ∈ V (c), we obtain I(u) ≥ m(c). Then I(un) = I(u) + I(wn) + on(1)→ m(c), which implies I(wn) ≤ on(1). Taking into account (3.14), we can see that ‖∇wn‖22 → 0. Therefore, un → u holds in H1(R3). Moreover, u is a minimizer for I on V (c). By Lemma 2.3, we know that all the minimizers of the functional I restricted on S(c) lie in Λ(c). We define m̄(c) := inf u∈Λ(c) I(u). EJDE-2023/56 CRITICAL SCHRÖDINGER-BOPP-PODOLSKY SYSTEM 17 By Lemma 3.5, I restricted to Λ(c) is bounded from below, which means that m̄(c) is well-defined. By an analogous argument, we can see that Proposition 2.7 also holds for m̄(c). This indicates that any minimizing sequence {ūn} on Λ(c) is relatively compact, i.e., ūn → ū in H1(R3). It is easily seen that {ūn} is a bounded Palais-Smale sequence for I on S(c), and thus ū is a ground state solution of (1.1) on S(c). � Acknowledgments. This work is supported by National Natural Science Foun- dation of China No. 11971095. References [1] D. G. Afonso, G. Siciliano; Normalized solutions to a Schrödinger-Bopp-Podolsky system under Neumann boundary conditions, Commun. Contemp. Math., 25 (2023), Paper No. 2150100, 20 pp. [2] B. Bartosz, J. 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Soc., 44 (2021), 953-980. Yuxin Li School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China Email address: liyx097@nenu.edu.cn Xiaojun Chang (corresponding author) School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, China Email address: changxj100@nenu.edu.cn Zhaosheng Feng School of Mathematical and Statistical Sciences, University of Texas Rio Grande Val- ley, Edinburg, TX 78539, USA Email address: zhaosheng.feng@utrgv.edu 1. Introduction 2. Preliminary results 3. Proof of Theorem ?? Acknowledgments References