Electronic Journal of Differential Equations, Vol. 2024 (2024), no. 43, pp. 1–25. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.43 GROUND STATE SOLUTIONS FOR NONLINEAR SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS WITH NONPERIODIC POTENTIALS QIAOYUN JIANG, LIN LI, SHANGJIE CHEN, GAETANO SICILIANO Abstract. In this article we study the existence of ground-state solutions for the Schrödinger-Bopp-Podolsky equations −∆u+ V (x)u+ ϕu = f(x, u) in R3 −∆ϕ+ a2∆2ϕ = 4πu2 in R3, where V ∈ C(R3,R) has different forms on the half spaces, i.e. V (x) = V1(x) for x1 > 0, and V (x) = V2(x) for x1 < 0, where V1, V2 ∈ C(R3) are periodic in each coordinate. The nonlinearity f is superlinear at infinity with subcritical or critical growth. 1. Introduction In this article we consider the existence of ground state solutions to Schrödinger- Bopp-Podolsky equations: −∆u+ V (x)u+ ϕu = f(x, u) in R3 −∆ϕ+ a2∆2ϕ = 4πu2 in R3 (1.1) where a > 0 is the Bopp-Podolsky (BP) parameter. This system, which was first studied in [13], appears when one looks for stationary solutions u(x)eiwt of the Schrödinger equation coupled with the Bopp-Podolsky Lagrangian of the electro- magnetic field. The Bopp-Podolsky theory, developed by Bopp [2], and independently by Podol- sky [3], is a second order theory for the electromagnetic field. As the Mie theory [21] and its generalizations given by Born and Infeld [4, 5, 6, 7], it was proposed to deal with the so called infinity problem that appears in the classical Maxwell the- ory. In fact, by the well-known Gauss law (or Poisson’s equation), the electrostatic potential ϕ for a given charge distribution whose density is ρ satisfies the equation −∆ϕ = ρ in R3. (1.2) 2020 Mathematics Subject Classification. 35B38, 35A15, 35Q55. Key words and phrases. Schrödinger-Bopp-Podolsky equation; variational method; Nehari manifold; critical growth. ©2024. This work is licensed under a CC BY 4.0 license. Submitted April 22, 2024. Published August 12, 2024. 1 2 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 If ρ = 4πδx0 , with x0 ∈ R3, the fundamental solution of (1.2) is G(x− x0), where G(x) = 1 |x| , and the electrostatic energy is EM (G) = 1 2 ∫ R3 |∇G|2 = +∞. Thus, to overcome this inconvenient new electromagnetic theories appeared. The most important ones are the Born-Infeld theory where equation (1.2) is replaced by −div ( ∇ϕ√ 1− |∇ϕ|2 ) = ρ in R3 and the Bopp-Podolsky theory where the equation for the electrostatic field is −∆ϕ+ a2∆2ϕ = ρ in R3. In both cases, if ρ = 4πδx0 , their solutions can be written explicitly, and the corresponding energy is finite. In this article, we focus on the Bopp-Podolsky theory, which then involves the study of the operator −∆+ a2∆2 whose fundamental solution satisfies −∆ϕ+ a2∆2ϕ = 4πδx0 and is given by K(x− x0), where K(x) := 1− e− |x| a |x| . In particular it presents no singularities at x0, since lim x→x0 K(x− x0) = 1 a and its energy is EBP(K) = 1 2 ∫ R3 |∇K|2 dx+ a2 2 ∫ R3 |∆K|2 dx <∞. We refer to [13] for more details. The most common Schrödinger-Bopp-Podolsky system is −∆u+ V (x)u+K(x)ϕu = f(x, u) in R3 −∆ϕ+ a2∆2ϕ = 4πu2 in R3 (1.3) In recent years, the question of the existence of solutions for (1.3) has been inten- sively studied by many researchers under a variety of conditions on V,K, f : we refer the reader to the papers [11, 13, 15, 16, 18, 22, 23, 27, 28] and the references therein. d’Avenia and Siciliano [13] firstly studied the system (1.3) where they assumed V (x) is a positive constant, K(x) = q2 and f(x, u) = |u|p−2u for p ∈ (2, 6). By using a suitable truncation and a useful splitting lemma, they obtained the existence and nonexistence of solutions. In particular, they take two different approaches to overcome the lack of compactness of the Sobolev embedding H1(R3) ↪→ Ls(R3), 2 < s < 6: by means of the Splitting Lemma and by looking for solutions in the subspace of radial functions ofH1(R3), both of which are only available for the case p ∈ (2, 6). EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 3 In [11, 18], the main results extended the existence results in [13] which only dealt with the subcritical case to critical case. In [11], by using some new analytic tech- niques and new inequalities, Chen and Tang assume K(x) = 1, f(x, u) = µg(u)+u5 and prove that system (1.3) admits ground state solutions for all µ > 0 if p ∈ (4, 6); for all µ > µ0 if p ∈ (2, 4]. In [18], Li, Pucci and Tang considered the system when K(x) = q2 and f(x, u) = µ|u|p−1u+ |u|4u. Under certain assumptions on V , they prove the existence of a nontrivial ground state solution, using the method of the Pohožaev-Nehari manifold, the arguments of Brezis-Nirenberg, the monotonicity trick and a global compactness lemma. Yang, Chen and Liu [28] assume V is coercive, K(x) = 1, f(x, u) = λg(u) + |u|4u. By using cut-off functions, the mountain pass theorem and Moser iteration, they prove the existence result without any growth and Ambrosetti-Rabinowitz conditions. Siciliano and Silva [23] assume V is a positive constant, K(x) = q2 and f(x, u) = |u|p−2u for p ∈ (2, 3]. Different from [13], they apply the fibering approach, and prove the system has no solutions at all for large values of q and has two radial solutions for small q. For the periodic potential and the nonperiodic potential, Yang, Yuan and Liu [27] study the existence of ground states for a nonlinear Schrödinger-Bopp-Podolsky system with asymptotically periodic potentials: V ∈ C(R3,R), 0 ≤ V (x) ≤ V∞(x) ∈ L∞(R3), for all x ∈ R3 and V − V∞ ∈ F . Here F = {k(x) : ∀ε > 0,m({x ∈ B1(y) : |k(x)| ≥ ε}) → 0 as |y| → ∞}. As a consequence, they also prove existence of ground states for the nonlinear Schrödinger-Bopp-Podolsky system with periodic potentials. In particular, Cheng and Wang [12] investigated the following Schrödinger- Poisson system with nonperiodic potential and subcritical exponent: −∆u+ V (x)u+ ϕu = a(x)|u|p−2u in R3 −∆ϕ = u2 in R3, (1.4) where p ∈ [4, 6), V, a ∈ C(R3,R), V (x) = { V1(x), x ∈ R3 +, V2(x), x ∈ R3 −, a(x) = { a1(x), x ∈ R3 +, a2(x), x ∈ R3 −. Here R3 ± = {x ∈ R3 : ±x1 > 0} and (H1) V1, V2, a1, a2 ∈ C(R3) are Tk periodic in the xk-direction for k = 1, 2, 3 with T1 = 1, (H2) essinf ai > 0, for i = 1, 2, (H3) min σ(−∆+ V ) > 0. Borrowing an idea from [14], they got a surface gap soliton ground state by using a variant of Lion concentration compactness lemma and based on the ground state energies of each periodic problem. Kang, Chen and Tang [17] investigated the following Schrödinger-Poisson system with nonperiodic potential and critical exponent: −∆u+ V (x)u+ ϕu = |u|4u+ λ|u|p−2u in R3 −∆ϕ = u2 in R3, (1.5) where p ∈ [4, 6), V belongs to C(R3,R) and satisfies the following assumptioons: 4 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 (A1) V0 = infx∈R3 V (x) > 0, and given V1 and V2 periodic functions in each coordinate direction, it is V (x) = { V1(x), x ∈ R3 +, V2(x), x ∈ R3 −. (A2) minσ(−∆+ V ) > 0. They prove the existence of ground state solutions by splitting lemma and some detailed analysis. For other papers about periodic and the nonperiodic potential, we refer to [1, 10, 19, 20, 24, 26, 30, 31] and the references therein. Motivated by the above works, we study a Schrödinger-Bopp-Podolsky system with nonperiodic potentials and subcritial and critical growth. First, we study the case of subcritial growth, i.e., −∆u+ V (x)u+ ϕu = |u|p−2u in R3 −∆ϕ+ a2∆2ϕ = 4πu2 in R3, (1.6) where p ∈ [4, 6), V belongs to C(R3,R) and satisfies (A1) and (A2). To state our results we need some preliminaries and notation, to used throughout this article. Let H1(R3) denote the usual Sobolev space with the standard scalar product and squared norm ∥u∥2H1 = ∫ R3 (|∇u|2 + u2) dx. When the domain of integration is not explicitly written, it is understood to be the whole space. We introduce the subspace of H1(R3), E := { u ∈ H1(R3) : ∫ V (x)u2 dx <∞ } which is a Hilbert space and ∥u∥2 := ∫ (|∇u|2 + V (x)u2)dx, ∀ u ∈ E. Assumption (A2) implies that ∥ · ∥H1 and ∥ · ∥ are two equivalent norms on E. Let Sq be the Sobolev embedding constant (see Theorem [25]), then ∥u∥q ≤ Sq∥u∥, ∀u ∈ E, 2 ≤ q ≤ 6. (1.7) Hereafter ∥ · ∥q is the norm in Lq(R3). Let D be the completion of C∞ c (R3) with respect to the norm ∥ · ∥D induced by the scalar product ⟨ϕ, ψ⟩ := ∫ (∇ϕ∇ψ + a2∆ϕ∆ψ) dx. Then D is a Hilbert space continuously embedded into D1,2(R3) and consequently in L6(R3). Fixed u ∈ E, the Lax-Milgram theorem [29] implies there exists a unique solution in D of the second equation in (1.6) and is given by ϕu(x) = K ∗ u2 = ∫ 1− e− |x−y| a |x− y| u2(y) dy. (1.8) Substituting (1.8) into the first equation of (1.6), we have −∆u+ V (x)u+ ϕuu = |u|p−2u in R3. (1.9) EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 5 Associated with (1.9) we have the energy functional I : E → R defined by I(u) = 1 2 ∫ (|∇u|2 + V (x)u2) dx+ 1 4 ∫ ϕuu 2 dx− 1 p ∫ |u|p dx. Furthermore, one can see that I is a C1 functional with the derivative given by ⟨I ′(u), v⟩ = ∫ (∇u∇v + V (x)uv) dx+ ∫ ϕuuv dx− ∫ |u|p−2uv dx, ∀u, v ∈ E. We define N := {u ∈ E : ⟨I ′(u), u⟩ = 0, u ̸= 0}, (1.10) which is the Nehari manifold of I. In this paper, we obtain the existence of ground state solution (1.6) by solving the minimization problem c := inf u∈N I(u). (1.11) By using Vi (i = 1, 2), we consider the auxiliary Schrödinger-Bopp-Podolsky system −∆u+ Vi(x)u+ ϕuu = |u|p−2u in R3. (1.12) Similarly, we define the working space Ei := { u ∈ H1(R3) : ∫ Vi(x)u 2 dx <∞ } , which is a Hilbert space and ∥u∥2Ei := ∫ R3 (|∇u|2 + Vi(x)u 2)dx, ∀ u ∈ Ei. By (A2), we have also minσ(∆ + Vi) > 0 for i = 1, 2. Then, we can deduce that ∥ · ∥, ∥ · ∥Ei are equivalent to ∥ · ∥H1 . Hence, ∥ · ∥ is equivalent to ∥ · ∥Ei , where i = 1, 2. In addition, the corresponding energy functional Ii : Ei → R is defined by Ii(u) = 1 2 ∫ (|∇u|2 + Vi(x)u 2) dx+ 1 4 ∫ ϕuu 2 dx− 1 p ∫ |u|p dx. It is a C1 functional with the derivative given by ⟨I ′ i(u), v⟩ = ∫ (∇u∇v + Vi(x)uv) dx+ ∫ ϕuuv dx− ∫ |u|p−2uv dx, ∀u, v ∈ Ei. The minimisation problem on the Nehari manifolds is Ni := {u ∈ Ei : ⟨I ′ i(u), u⟩ = 0, u ̸= 0} and ci := inf u∈Ni Ii(u). (1.13) Now, we summarize our first results as follows. Theorem 1.1. Suppose (A1) and (A2) hold and p ∈ [4, 6). If c < min{c1, c2}, then (1.6) has a positive ground state solution u0 with I(u0) = c. A sufficient condition that guarantees c < min{c1, c2} is given in the next result. Theorem 1.2. Suppose (A1) and (A2) hold and p ∈ [4, 6). Let wi be a positive ground state solution to (1.12) for i = 1, 2 and assume that either c1 ≤ c2, ∫ R3 − (V2 − V1)w 2 1 dx < 0, or c2 ≤ c1, ∫ R3 + (V1 − V2)w 2 2 dx < 0. 6 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 Then, c < min{c1, c2} and thus (1.6) has a positive ground state solution. Secondly, we study the case of critial growth. −∆u+ V (x)u+ ϕu = |u|4u+ λ|u|p−2u in R3 −∆ϕ+ a2∆2ϕ = 4πu2 in R3, (1.14) where p ∈ [4, 6), λ > 0, V (x) ∈ C(R3,R) and satisfies (A1), (A2). As before we define the main objects. Similar to the case of subcritial growth, we define the energy functional J (u) = 1 2 ∫ (|∇u|2 + V (x)u2) dx+ 1 4 ∫ ϕuu 2 dx− 1 6 ∫ |u|6 dx− λ p ∫ |u|p dx, and for all u, v ∈ E, its derivative satisfies ⟨J ′(u), v⟩ = ∫ (∇u∇v+V (x)uv) dx+ ∫ ϕuuv dx− ∫ |u|4uv dx−λ ∫ |u|p−2uv dx. We define the Nehari manifold of J and the minimization problem M := {u ∈ E : ⟨J ′(u), u⟩ = 0, u ̸= 0} and m := inf u∈M J (u). (1.15) By using Vi, we consider the auxiliary Schrödinger-Bopp-Podolsky system −∆u+ Vi(x)u+ ϕuu = |u|4u+ λ|u|p−2u in R3. (1.16) For all u, v ∈ Ei, we have Ji(u) = 1 2 ∫ (|∇u|2 + Vi(x)u 2) dx+ 1 4 ∫ ϕuu 2 dx− 1 6 ∫ |u|6 dx− λ p ∫ |u|p dx, ⟨J ′ i (u), v⟩ = ∫ (∇u∇v + Vi(x)uv) dx+ ∫ ϕuuv dx− ∫ |u|4uv dx− λ ∫ |u|p−2uv dx. Let the Nehari manifold of Ji be Mi := {u ∈ Ei : ⟨J ′ i (u), u⟩ = 0, u ̸= 0} and mi := inf u∈Mi Ji(u). (1.17) We have our second result. Theorem 1.3. Suppose (A1) and (A2) hold, m < min{m1,m2} and either p ∈ (4, 6), λ > 0 or p = 4, λ > 0 sufficiently large. Then (1.14) has a positive ground state solution û0 with J (û0) = m. Next we give a condition that guarantees m < min{m1,m2}. Theorem 1.4. Suppose (A1) and (A2) hold and p ∈ [4, 6). Let vi be a positive ground state solution to (1.16) for i = 1, 2 and either m1 ≤ m2, ∫ R3 − (V2 − V1)v 2 1 dx < 0, or m2 ≤ m1, ∫ R3 + (V1 − V2)v 2 2 dx < 0. Then, m < min{m1,m2} and thus (1.14) has a positive ground state solution. EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 7 Notation. We use following notation along this article. • C, C̄ and Ci (i = 1, 2, . . . ) denote positive constants which may change from line to line. • → and ⇀ denote strong and weak convergence in the related function spaces, respectively. • ̸→ and ̸⇀ denote not strong and weak convergence in the related function spaces, respectively. • BR(x0) denotes the ball centered at x0 ∈ R3 with radius R. • p′ = p p−1 is the conjugate exponent of p, E−1 denotes the dual space of E. • on(1) denotes a vanishing sequence in the specified space. • S is the best Sobolev constant for the embedding of D1,2(R3) in L6(R3). 2. Preliminary results In this section, we give some properties of ϕu, which will be used later. Lemma 2.1 ([13, Lemma 3.4]). For every u ∈ H1(R3) we have: (i) for every y ∈ R3, ϕu(·+y) = ϕu(·+ y); (ii) ϕu ≥ 0; (iii) for every s ∈ (3,+∞], ϕu ∈ Ls(R3) ∩ C0(R3); (iv) for every s ∈ ( 32 ,+∞],∇ϕu ∈ Ls(R3) ∩ C0(R3); (v) ϕu ∈ D; (vi) ∥ϕu∥6 ≤ C∥u∥2; (vii) ϕu is the unique minimizer in D of the functional E(ϕ) = 1 2 ∥∇ϕ∥22 + a2 2 ∥∆ϕ∥22 − ∫ ϕu dx, ∀ϕ ∈ D; (viii) ∫ ϕuu 2dx ≤ S2∥u∥412/5; (ix) if un ⇀ u in H1(R3), then ϕun ⇀ ϕu in D. Let us define the function Ψ : H1(R3) → R by Ψ(u) = ∫ ϕuu 2 dx. It is clear that for all fixed u ∈ H1(R3), we have Ψ(u(·+ y)) = Ψ(u) for all y ∈ R3 and that Ψ is weakly lower semi-continuous in H1(R3). The next lemma shows that the functional Ψ and its derivative Ψ′ have the Brezis-Lieb splitting property, which is similar to the well-known Brézis-Lieb lemma. Lemma 2.2 ([18, Lemma 2.5]). If un ⇀ u in E and un → u a.e. in R3, then (i) Ψ(un − u) = Ψ(un)−Ψ(u) + on(1). (ii) Ψ′(un − u) = Ψ′(un)−Ψ′(u) + on(1) in E −1. Lemma 2.3 (Brézis-Lieb lemma [8]). If un ⇀ u in E, then: (i) ∥un − u∥2 = ∥un∥2 − ∥u∥2 + on(1). (ii) ∥un − u∥ss = ∥un∥ss − ∥u∥ss + on(1), where s ∈ (2, 6]. (iii) ∥un − u∥s−2(un − u) = ∥un∥s−2un − ∥u∥s−2u+ on(1) in E −1. 8 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 3. Subcritical case In this section, we give the proof of Theorems 1.1 and 1.2. First, we give some properties of N defined in (1.10). Lemma 3.1. Suppose that (A1), (A2) are satisfied, then we have: (i) for any u ∈ E \ {0}, there exists a unique tu > 0 such that tuu ∈ N . Moreover, I(tuu) = maxt>0 I(tu); (ii) N is a natural constraint for the functional I, i.e., critical points of I on N are critical points of I on E; (iii) the functional I is bounded away from zero on N , i.e., c = infu∈N I(u) > 0. Proof. (i) We first consider the case p > 4. In fact, for all t > 0, tu ∈ N is equivalent to t2∥u∥2 + t4 ∫ ϕuu 2 dx = tp ∫ |u|p dx. Set a1 = ∥u∥2 > 0, a2 = ∫ ϕuu 2 dx > 0, a3 = ∫ |u|p dx > 0. Then we obtain a1t 2 + a2t 4 = a3t p. Let g(t) = a1t 2 + a2t 4 − a3t p. Since p > 4, then g(t) → −∞ as t → ∞ and g(t) > 0 as t → 0. So there exists a solution t = tu > 0 such that g(t) = 0, i.e., tuu ∈ N . Furthermore, since I ′(tu) = g(t), we deduce that I(tuu) = maxt>0 I(tu). It remains to show the uniqueness of tu. In fact, g′(t) = 2a1t+ 4a2t 3 − pa3t p−1, g′′(t) = 2a1 + 12a2t 2 − p(p− 1)a3t p−2, g′′′(t) = 24a2t− p(p− 1)(p− 2)a3t p−3, By a direct calculation, we obtain that g′′′(t) = 0 has a unique solution t′′′u , and g′′′(t) > 0 with 0 < t < t′′′u , g′′′(t) < 0 with t > t′′′u . So g′′(t) = 0 has a unique solution t′′u, and g′′(t) > 0 with 0 < t < t′′u, g ′′(t) < 0 with t > t′′u. By iterating this procedure, we obtain the uniqueness of tu. Next, we consider the case p = 4. Define A = {u ∈ E \ {0} : ∫ ϕuu 2 dx < ∫ u4 dx}. We show that the set A is non-empty. In fact, take u0 ∈ C∞ 0 (R3, [0, 1]) satisfying u0 = 1 in BR0 (0) and u0 = 0 for R3 \ B2R0 (0), where R0 is a positive constant to be determined. Then by Lemma 2.1, on one hand,∫ ϕu0 u20 dx ≤ S2 (∫ B2R0 (0) |u0|12/5 dx )5/3 ≤ S2 4 3 π(2R0) 5. (3.1) On the other hand, ∫ u40 dx ≥ ∫ BR0 (0) u40 dx = 4 3 πR3 0, (3.2) and a suitable choice of R0 permits to have S2(2R0) 5 < R3 0, implying that u0 ∈ A. Let h(t) = I(tu) = t2 2 ∥u∥2 + t4 4 ( ∫ ϕuu 2 dx− ∫ u4 dx). We take u ∈ A, it is easy to verify that h(t) > 0 for t sufficient small and h(t) < 0 for t sufficient large. Then similar to the case of p ∈ (4, 6), it is not difficult to EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 9 verify that there exists a unique tu > 0 such that h′(tu) = 0, i.e., tuu ∈ N and I(tuu) = maxt>0 I(tu). (ii) For each u ∈ N , from (1.7) and (1.10), 0 = ∥u∥2 + ∫ ϕuu 2 dx− ∫ |u|p dx ≥ ∥u∥2 − ∫ |u|p dx ≥ ∥u∥2 − Sp p∥u∥p. Since p ∈ [4, 6), this implies that ∥u∥ ≥ ( 1 Sp p ) 1 p−2 > 0. (3.3) Define A(u) := ⟨I ′(u), u⟩, by a direct computation, ⟨A′(u), u⟩ = 2∥u∥2 + 4 ∫ ϕuu 2 dx− p ∫ |u|p dx = (2− p)∥u∥2 + (3− p) ∫ ϕuu 2 dx ≤ (2− p)∥u∥2 < (2− p) ( 1 Sp p ) 1 p−2 < 0. (3.4) Then there exists µ ∈ R such that I ′(u) = µA′(u). Therefore 0 = ⟨I ′(u), u⟩ = µ⟨A′(u), u⟩, which implies µ = 0 by (3.4) and then I ′(u) = µA′(u) = 0. (iii) For any u ∈ N , we can deduce from p ∈ [4, 6) and (3.3) that I(u) = ( 1 2 − 1 p )∥u∥2 + ( 1 4 − 1 p ) ∫ ϕuu 2 dx ≥ ( 1 2 − 1 p )∥u∥2 ≥ ( 1 2 − 1 p )( 1 Sp p ) 2 p−2 > 0. This completes the proof. □ Corollary 3.2. For any u ∈ Ei \ {0}, there exists a unique tui > 0 such that tuiu ∈ Ni. Moreover, Ii(tuiu) = maxt>0 Ii(tu), where i = 1, 2. Lemma 3.3. Suppose that (A1), (A2) are satisfied, then there exists a (PS)c se- quence {un} ⊂ N , namely such that I(un) → c and I ′(un) → 0 as n → +∞. Moreover, the sequence {un} is bounded and bounded away from zero on N . Proof. By the Ekeland Variational Principle, there exists a sequence {un} ⊂ N such that I(un) → c and I ′|N (un) → 0 as n→ +∞. Since un ∈ N , we obtain 0 = ⟨I ′(un), un⟩ = ∥un∥2 + ∫ ϕun u2n dx− ∫ |un|p dx, and by a direct calculation, for p ∈ [4, 6), I(un) = I(un)− 1 4 ⟨I ′(un), un⟩ = 1 4 ∥un∥2 + ( 1 4 − 1 p ) ∫ |un|p dx ≥ 1 4 ∥un∥2. 10 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 Then, it follows from I(un) → c as n → +∞ that {un} is bounded. From the definition of A in the proof of Lemma 3.1, we have as n→ +∞, on(1) = ∇|N I(un) = I ′(un) + µnA′(un) (3.5) for some µn ∈ R. Taking the scalar product with un (which is bounded), we obtain that on(1) = ⟨I ′(un), un⟩+ µn⟨A′(un), un⟩ = µn⟨A′(un), un⟩. (3.6) Since un ∈ N and p ∈ [4, 6), we obtain, as in equation (3.4) ⟨A′(un), un⟩ < (2− p) ( 1 Sp p ) 1 p−2 < 0, and (3.6) gives µn → 0. It follows from (3.5) and (3.6) that µn → 0 and I ′(un) → 0 in E−1 as n→ +∞. Moreover, by (3.3), {un}n∈N is bounded and bounded away from zero. □ Since {un} is bounded in E, passing to a subsequence, there exists u ∈ E such that as n→ ∞, un ⇀ u in E, un → u in Lr loc (R3), r ∈ [1, 6), un(x) → u(x) a.e. on R3. (3.7) Here is some preparatory work that will be used later. Lemma 3.4 ([13, Lemma B.5]). Let {yn} ⊂ R3, v ∈ H1(R3), {vn} ⊂ H1(R3) be bounded. (i) If |yn| → +∞, then v(·+ yn)⇀ 0 in H1(R3). (ii) If {yn} is bounded, then, up to a subsequence, vn ̸⇀ 0 in H1(R3) =⇒ vn(·+ yn) ̸⇀ 0 in H1(R3) Now, we give the splitting lemma of a (PS)c sequence of I, which plays a crucial role for the subsequent discussion. Lemma 3.5 (Splitting Lemma). Let {un} ⊂ E be a bounded (PS)c sequence of I at level c > 0 and assume that un ⇀ u in E. Then, passing to a subsequence, either un strongly converges to u, or setting k ∈ N∪{0}, there are sequences {ui}ki=1 ⊂ E and yin ∈ T1Z× T2Z× T3Z ⊂ R3 with 1 ≤ i ≤ k such that (i) (−1)εi−1(yin)1 → +∞ and |(yin)1 − (yjn)1| → +∞ for 1 ≤ i ̸= j ≤ k, as n→ +∞, where (yin)γ denotes the γ-th component of yin, 1 ≤ γ ≤ 3. (ii) un → u+ ∑k i=1 u i(· − yin) in E. (iii) I(un) = I(u) + ∑k i=1 Iεi(ui) + on(1). (iv) I ′ εi(u i) = 0 and ui ̸= 0 with 1 ≤ i ≤ k. where εi = {1, 2}. Proof. Let us divide the proof in various steps. Step 1: Let z1n = un − u. We have two possibilities: If z1n → 0 in E, then the first alternative follows and the proof is concluded. If z1n ̸→ 0 in E. Let εi = {1, 2} and R3 (−1)εi = { R3 −, εi = 1, R3 +, εi = 2. (3.8) EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 11 Since z1n ⇀ 0 in E, it follows from Lemma 2.2 and Lemma 2.3, for any φ ∈ C∞ 0 (E,R), as n→ ∞, ⟨I ′(un), φ⟩ = ∫ (∇un∇φ+ V (x)unφ) dx+ ∫ ϕun unφdx− ∫ |un|p−2unφdx = ∫ (∇u∇φ+ V (x)uφ) dx+ ∫ (∇z1n∇φ+ V (x)z1nφ) dx+ ∫ ϕuuφdx + ∫ ϕz1 n z1nφdx− ∫ |u|p−2uφdx− ∫ |z1n|p−2z1nφdx+ on(1) = ⟨I ′(u), φ⟩+ ⟨I ′(z1n), φ⟩+ on(1) = ⟨I ′(u), φ⟩+ ∫ (∇z1n∇φ+ V1z 1 nφ) dx+ ∫ R3 − V2z 1 nφdx + ∫ ϕz1 n z1nφdx− ∫ |z1n|p−2z1nφdx+ on(1) = ⟨I ′(u), φ⟩+ ⟨I ′ ε1(z 1 n), φ⟩+ ∫ R3 (−1)ε1 (−1)ε1(V1 − V2)z 1 nφdx+ on(1) = ⟨I ′(u), φ⟩+ ⟨I ′ ε1(z 1 n), φ⟩+ on(1). Which together with I ′(un) → 0 and I ′(u) = 0 imply that ⟨I ′ ε1(z 1 n), φ⟩ → 0, as n→ ∞. (3.9) In addition, for any φ ∈ C∞ 0 (E,R), by Lemmas 2.2 and 2.3, as n→ ∞, ⟨I ′(un), φ⟩ = ⟨I ′(u), φ⟩+ ⟨I ′(z1n), φ⟩+ on(1). (3.10) Since I ′(un) → 0 as n→ ∞, From (3.10) and I ′(u) = 0 it follows that ⟨I ′(z1n), φ⟩ → 0. Setting φ = z1n, we obtain ⟨I ′(z1n), z 1 n⟩ → 0, i.e., ∥z1n∥2 + ∫ ϕz1 n (z1n) 2 dx− ∫ |z1n|p dx→ 0. (3.11) Let δ1 := lim sup n→∞ sup y∈R3 ∫ B1(y) |z1n|2 dx. We claim that δ1 ̸= 0. If not, by Lion’s lemma, z1n → 0 in Lr(R3) for r ∈ (2, 6). By Hölder’s inequality, we obtain that∫ ϕz1 n (z1n) 2 dx ≤ (∫ |ϕz1 n |6 dx )1/6(∫ |z1n|12/5 dx )5/6 ≤ CS∥z1n∥212/5 → 0. From (3.11) it follows that ∥z1n∥2 → 0, which contradicts z1n ̸→ 0 in E. Then, there exists y1n ∈ T1Z× T2Z× T3Z ⊂ R3 such that∫ B1(y1 n) |z1n|2dx ≥ δ1 2 . (3.12) Let ξ1n := (0, (y1n)2, (y 1 n)3), σ 1 n := ((y1n)1, 0, 0), and w 1 n = z1n(·+ξ1n). Clearly, ∥∥w1 n ∥∥ =∥∥z1n∥∥ and w1 n ⇀ 0 in E but not strongly. Therefore, by (3.12), we obtain∫ B1(σ1 n) |w1 n|2dx ≥ δ1 2 . (3.13) 12 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 It is easy to check that |(y1n)1| = |σ1 n| → +∞, that is, (−1)ε1−1(y1n)1 → +∞. (3.14) Considering the sequence {w1 n(·+ σ1 n)}, which is bounded in Eε1 , there exists u1 ∈ Eε1 satisfying w1 n(·+ σ1 n)⇀ u1 in Eε1 , w1 n(·+ σ1 n) → u1 in Lr loc (R3), w1 n(x+ σ1 n) → u1(x) a.e. in R3. From (3.13), we obtain u1 ̸= 0. From (3.9), for any φ ∈ C∞ 0 (E,R), we obtain ⟨I ′ ε1(w 1 n(·+ σ1 n)), φ⟩ = ⟨I ′ ε1(z 1 n(·+ y1n)), φ⟩ = ⟨I ′ ε1(z 1 n), φ(· − y1n)⟩ → 0. Hence, we have I ′ ε1(w 1 n(·+ σ1 n)) → 0 since C∞ 0 (E,R) is dence in E, it follows that I ′ ε1(u 1) = 0. (3.15) Step 2: Let z2n = z1n − u1(x − y1n). Then, z2n ⇀ 0 in E because the norm of E is equivalent to the norm of Eε1 . In addition, from Lemmas 2.2 and 2.3, by the simple calculation, as n→ ∞, I(un) = 1 2 ∥un∥2 + 1 4 ∫ ϕun u2n dx− 1 p ∥un∥pp = 1 2 (∥z1n∥2 + ∥u∥2) + 1 4 (∫ ϕuu 2 dx+ ∫ ϕz1 n (z1n) 2 dx ) − 1 p (∥z1n∥pp + ∥u∥pp) + on(1) = I(z1n) + I(u) + on(1) = I(z2n) + I(u1) + I(u) + on(1) = I(z2n) + Iε1(u1) + ∫ R3 (−1)ε1 (−1)ε1(V1 − V2)u 1 dx+ I(u) + on(1) = I(z2n) + Iε1(u1) + I(u) + on(1). (3.16) We have two possibilities: If z2n → 0 in E, i.e., z1n − u1(x− y1n) = un − u− u1(x− y1n) → 0, i.e., un → u+ u1(x− y1n). (3.17) From (3.16), we obtain I(un) = I(u) + Iε1(u1) + on(1). (3.18) Then the Lemma is proved for k = 1. It follows from (3.14), (3.17), (3.18), and (3.15). EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 13 If z2n ̸→ 0 in E. From Lemmas 2.2 and 2.3, we obtain that for any φ ∈ C∞ 0 (E,R), as n→ ∞, ⟨I ′(z1n), φ⟩ = ⟨I ′(z2n), φ⟩+ ⟨I ′(u1), φ(x+ y1n)⟩+ on(1) = ⟨I ′(z2n), φ⟩+ ⟨I ′ ε1(u 1), φ(x+ y1n)⟩ + ∫ R3 (−1)ε1 (−1)ε1(V1 − V2)u 1φ(x+ y1n)dx+ on(1) = ⟨I ′(z2n), φ⟩+ ⟨I ′ ε1(u 1), φ(x+ y1n)⟩+ on(1). (3.19) It follows from ⟨I ′(z1n), φ⟩ → 0 and I ′ ε1(u 1) = 0 that ⟨I ′(z2n), φ⟩ → 0. Setting φ = z2n, we obtain ⟨I ′(z2n), z 2 n⟩ → 0, i.e., ∥z2n∥2 + ∫ ϕz2 n (z2n) 2dx− ∫ |z2n|pdx→ 0. Let δ2 := lim sup n→∞ sup y∈R3 ∫ B1(y) |z2n|2dx. Simlar to δ1 ̸= 0, we obtain δ2 ̸= 0. Then, there exists y2n ∈ T1Z× T2Z× T3Z ⊂ R3 such that ∫ B1(y2 n) |z2n|2dx ≥ δ2 2 . Let ξ2n := (0, (y2n)2, (y 2 n)3), σ 2 n := ((y2n)1, 0, 0), and w 2 n = z2n(x+ξ 2 n). Clearly, ∥w2 n∥ = ∥z2n∥ and w2 n ⇀ 0 in E. Therefore,∫ B1(σ2 n) |w2 n|2dx ≥ δ2 2 . (3.20) It is easy to check that as n→ ∞, |(y2n)1| = |σ2 n| → +∞; that is, (−1)ε2−1(y2n)1 → +∞. (3.21) Then w2 n(·+ σ2 n) ̸⇀ 0 in E. In addition, we claim that |(y2n)1 − (y1n)1| → +∞. (3.22) To see this, first observe that w2 n(·+ σ2 n + (y2n)1 − (y1n)1) = z2n(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n) = z1n(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n)− u1(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n − y1n) = w1 n(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n − ξ1n)− u1(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n − y1n) = w1 n(·+ σ2 n + y2n − y1n)− u1(·+ σ2 n + y2n − y1n − σ1 n) From Lemma 3.4 (ii), since w2 n(·+ σ2 n) ̸⇀ 0 in E, if it were |(y2n)1 − (y1n)1| ̸→ +∞, we obtain w2 n(·+ σ2 n + (y2n)1 − (y1n)1) ̸⇀ 0. On the other hand, since w1 n(·+ σ1 n)⇀ u1, we obtain w1 n(·+ σ2 n + y2n − y1n)− u1(·+ σ2 n + y2n − y1n − σ1 n)⇀ 0, 14 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 which is a contradiction. So we obtain (3.22). Moreover, for any φ ∈ C∞ 0 (E,R), we obtain as n→ ∞, ⟨I ′ ε2(z 1 n), φ⟩ = ⟨I ′ ε2(z 2 n), φ⟩+ ⟨I ′ ε2(u 1), φ(x+ y1n)⟩ = ⟨I ′ ε2(z 2 n), φ⟩+ ⟨I ′ ε1(u 1), φ(x+ y1n)⟩ + ∫ (Vε2 − Vε1)u 1φ(x+ y1n)dx+ on(1) = ⟨I ′ ε2(z 2 n), φ⟩+ ⟨I ′ ε1(u 1), φ(x+ y1n)⟩+ on(1). (3.23) By ⟨I ′ ε2(z 1 n), φ⟩ → 0 and I ′ ε1(u 1) = 0, we obtain I ′ ε2(z 2 n) → 0 since C∞ 0 (E,R) is dense in E. Considering the sequence {w2 n(·+σ2 n)}, which is bounded in Eε2 , there exists u2 ∈ Eε2 satisfying w2 n(·+ σ2 n)⇀ u2 in Eε2 , w2 n(·+ σ2 n) → u2 in Lr loc (R3), w2 n(x+ σ2 n) → u2(x) a.e. on R3. We can see that u2 ̸= 0. For any φ ∈ C∞ 0 (E,R), we obtain ⟨I ′ ε2(u 2), φ⟩ = ⟨I ′ ε2(w 2 n(·+ σ2 n)), φ⟩+ on(1) = ⟨I ′ ε2(z 2 n(·+ y2n)), φ⟩+ on(1) = ⟨I ′ ε2(z 2 n), φ(· − y2n)⟩+ on(1) = on(1). Hence, since C∞ 0 (E,R) is dense in E, we have I ′ ε2(u 2) = 0. (3.24) Step 3: Let z3n = z2n − u2(x− y2n). From (3.16), we have I(un) = I(z2n) + Iε1(u1) + I(u) + on(1) = I(z3n) + Iε2(u2) + Iε1(u1) + I(u) + on(1). (3.25) We have two possibilities: If z3n → 0 in E, i.e., z2n−u2(x− y2n) = un−u−u1(x− y1n)−u2(x− y2n) → 0, i.e., un → u+ u1(x− y1n) + u2(x− yn). (3.26) From (3.25), we obtain I(un) = I(u) + Iε1(u1) + Iε2(u2) + on(1). (3.27) Then the Lemma is proved for k = 2. It follows from (3.21), (3.22), (3.26), (3.27), and (3.24). If z2n ̸→ 0 in E, we just repeat the argument. Step k: By I(un) = I(u)+ ∑k i=1 Iεi(ui)+ on(1), since Iεi(ui) ≥ cεi ≥ min{c1, c2} and I(un) is bound, the iteration must stop at some finite index k. The proof is complete □ Proof of Theorem 1.1. In view of Lemma 3.3, we obtained that there exists a bounded (PS)c sequence {uk} ⊂ N such that I(uk) → c and I ′(uk) → 0 as k → +∞. Since {uk} is bounded in E, the, going to a subsequence if necessary, still denoted by {uk}, we can suppose that there exists u ∈ E such that uk ⇀ u in EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 15 E. With c < min{c1, c2}, by Lemma 3.4, if uk ̸→ u, we can obtain that k ≥ 1 and nontrivial solutions u1, u2, . . . , uj of Iεj with εj = {1, 2} satisfy c = lim k→+∞ I(uk) = I(u) + k∑ j=1 Iεj (uj) ≥ kmin{c1, c2} ≥ min{c1, c2}, which is contradiction with c < min{c1, c2}. Thus, uk → u, and then I(u) = c and I ′(u) = 0. Obviously, u ̸= 0. Therefore, u is a ground state solution of (1.6). Considering u0 = |u|, it is easy to check that I(u0) = I(u) = c and u0 ∈ N . From standard arguments, we infer that I ′(u0) = 0. Thus, ū0 is a non-negative solution of system (1.6). Furthermore, the strong maximum principle implies that u0 > 0 in R3, and thus, u0 is a positive ground state solution of system (1.6). □ Proof of Theorem 1.2. We just study the case of c1 ≤ c2. The case c2 ≤ c1 is analogous. Let w1 ∈ N1 ⊂ E be a positive ground state for the purely periodic problem for (1.6) with i = 1. We can see from Lemma 3.1 that there exists s > 0 satisfying sw1 ∈ N . Then, from the assumption of Theorem 1.2 and Corollary 3.2, we have c ≤ I(sw1) = s2 2 ∥w1∥2 + s4 4 ∫ ϕw1 |w1|2dx− sp p ∥w1∥pp = s2 2 ∥w1∥2E1 + s4 4 ∫ ϕw1 |w1|2 dx− sp p ∥w1∥pp + s2 2 ∫ R3 − (V2 − V1)w 2 1 dx = I1(sw1) + s2 2 ∫ R3 − (V2 − V1)w 2 1 dx < I1(w1) = c1, which implies that c < c1. Thus, c < min{c1, c2}. □ 4. Critical case In this section, we give the proof of Theorems 1.3 and 1.4. According to [24, Theorem 1.2], we have the following result. Theorem 4.1. Assume that (A1) holdls and minσ(−∆+Vi) > 0. Then, for every λ > 0 and p ∈ (4, 6), (1.16) has a positive ground state solution. Moreover, if p = 4, then (1.16) possesses positive ground state solutions for λ > 0 sufficiently large. Similar to the case of subcritical, we have some properties of M . Lemma 4.2. Suppose that (A1), (A2) are satisfied, then we have (i) For any u ∈ E \ {0}, there exists a unique t̂u > 0 such that t̂uu ∈ M. Moreover, J (t̂uu) = maxt>0 I(tu). (ii) M is a natural constraint for the functional J , i.e., critical points of J on M are critical points of J on E. (iii) The functional J is bounded away from zero on M, i.e., m = infu∈M J (u) > 0. Proof. (i) For t > 0, tu ∈ M is equivalent to t2∥u∥2 + t4 ∫ ϕuu 2 dx = t6∥u∥66 + λtp∥u∥pp. 16 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 Set a1 = ∥u∥2 > 0, a2 = ∫ ϕuu 2 dx > 0, a3 = ∥u∥66 > 0, a4 = ∥u∥pp > 0. Then we obtain a1t 2 + a2t 4 = a3t 6 + λa4t p. Let ĝ(t) = a1t 2 + a2t 4 − a3t 6 − λa4t p. Since p > 4, then ĝ(t) → −∞ as t → ∞ and ĝ(t) > 0 as t → 0. So there exists a solution t̂ = t̂u > 0 such that g(t̂) = 0, i.e., t̂uu ∈ M. Furthermore, since J ′(tu) = ĝ(t), we deduce that J (t̂uu) = maxt>0 J (tu). It remains to show the uniqueness of t̂u. In fact, suppose by contradiction that there exists 0 < t1 < t2 such that ĝ(t1) = ĝ(t2) = 0. Then ∥u∥2 t21 + ∫ ϕuu 2 dx = t21∥u∥66 + λtp−4 1 ∥u∥pp, ∥u∥2 t22 + ∫ ϕuu 2 dx = t22∥u∥66 + λtp−4 2 ∥u∥pp. As a consequence,( 1 t22 − 1 t21 ) ∥u∥2 = (t22 − t21)∥u∥66 + λ(tp−4 2 − tp−4 1 )∥u∥pp, which is impossible by 0 < t1 < t2. (ii) For any u ∈ M, we have that 0 = ∥u∥2 + ∫ ϕuu 2 dx− ∥u∥66 − λ∥u∥pp ≥ ∥u∥2 − ∥u∥66 − λ∥u∥pp ≥ ∥u∥2 − S6 6∥u∥6 − λSp p∥u∥p. Since p ∈ [4, 6), there exists a constant Λ0 > 0 such that ∥u∥ ≥ Λ0 > 0. (4.1) We define B(u) := ⟨J ′(u), u⟩, by a direct computation, ⟨B′(u), u⟩ = 2∥u∥2 + 4 ∫ ϕuu 2 dx− 6∥u∥66 − λp∥u∥pp = (2− p)∥u∥2 + (4− p) ∫ ϕuu 2 dx+ (p− 6)∥u∥66 ≤ (2− p)∥u∥2 < 0. (4.2) Then there exists µ̂ ∈ R such that J ′(u) = µ̂B′(u). Therefore 0 = ⟨J ′(u), u⟩ = µ̂⟨B′(u), u⟩, which implies µ̂ = 0 by (4.2) and then J ′(u) = µ̂B′(u) = 0. (iii) For any u ∈ M, we can deduce from p ∈ [4, 6) and (4.1) that J (u) = ( 1 2 − 1 p )∥u∥2 + ( 1 4 − 1 p ) ∫ ϕuu 2 dx+ ( 1 p − 1 6 )∥u∥66 ≥ ( 1 2 − 1 p )∥u∥2 ≥ ( 1 2 − 1 p )Λ2 0 > 0. This completes the proof. □ EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 17 Corollary 4.3. For any u ∈ Ei \ {0}, there exists a unique t̂ui > 0 such that t̂uiu ∈ Mi. Moreover, Ji(t̂uiu) = maxt>0 Ji(tu), where i = 1, 2. Lemma 4.4. Suppose that (A1), (A2) are satisfied, then there exists a (PS)c se- quence {un} ⊂ M such that J (un) → c and J ′(un) → 0 as n → +∞. Moreover, the sequence {un} is bounded and bounded away from zero on N . Proof. By the Ekeland Variational Principle, there exists a sequence {un} ⊂ M such that J (un) → c and J ′|M(un) → 0 as n→ +∞. From the definition of B in the proof of Lemma 4.2, we have, as n→ +∞, on(1) = ∇|MJ (un) = J ′(un) + µ̂nB′(un), (4.3) for some µ̂n ∈ R. Taking the scalar product with uk, we obtain that on(1) = ⟨J ′(un), un⟩+ µ̂n⟨B′(un), un⟩ = µ̂n⟨B′(un), un⟩. (4.4) Since un ∈ M and p ∈ [4, 6), we obtain ⟨B′(un), un⟩ = 2 ∥un∥2 + 4 ∫ ϕunu 2 n dx− 6 ∫ |un|6 dx− λp ∫ |un|p dx = (2− p) ∥un∥2 + (4− p) ∫ ϕun u2n dx+ (p− 6) ∫ |un|6 dx ≤ (2− p) ∥un∥2 < 0. It follows from (4.3) and (4.4) that µ̂n → 0 and J ′(un) → 0 in E−1 as n → +∞. It remains to show that {un} is bounded in E. Since un ∈ M, we obtain 0 = ⟨J ′(un), un⟩ = ∥un∥2 + ∫ ϕun u2n dx− ∫ |un|6 dx− λ ∫ |un|p dx. By a direct calculation, for p ∈ [4, 6), J (un) = J (un)− 1 4 ⟨J ′(un), un⟩ = 1 4 ∥un∥2 + 1 12 ∫ |un|6 dx+ λ( 1 4 − 1 p ) ∫ |un|p dx ≥ 1 4 ∥un∥2. Then, it follows from J (un) → c as n→ +∞ that {un} is bounded. Moreover, by (4.1), {un} is bounded and bounded away from zero. □ Since {un} is bounded in E, passing to a subsequence, there exists û ∈ E such that as n→ ∞, un ⇀ û in E, un → û in Lr loc (R3), r ∈ [1, 6), un(x) → û(x) a.e. on R3. (4.5) For ε > 0, let φε(x) := 31/4ψ(x)ε1/2 (ε2 + |x|2)1/2 , 18 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 where ψ ∈ C∞ 0 (R3, [0, 1]) is such that ψ(x) = 1 for |x| ≤ R and ψ(x) = 0 for |x| ≥ 2R. We need the following asymptotic estimates as ε→ 0+ (see [9]) ∥∇φε∥22 = S3/2 +O(ε), ∥φε∥66 = S3/2 +O(ε3) ∥φε∥ss =  O(εs/2), if s ∈ [2, 3) O(εs/2| ln ε|), if s = 3, O(ε(6−s)/2), if s ∈ (3, 6) (4.6) Lemma 4.5. Suppose that (A1) and (A2) are satisfied. Then 0 < mi < 1 3 S3/2, i = 1, 2, (4.7) where mi defined in (1.17), if one of the following conditions is satisfied: (i) 4 < p < 6 and λ > 0; (ii) p = 4 and λ > 0 large enough. Proof. From the definition of mi and Lemma 4.2, there exists tε > 0 such that 0 < mi ≤ Ji(tεφε) = max t≥0 Ji(tφε). (4.8) On the one hand, since 0 is a local minimum of Ji, there exists a constant C > 0, independent of ε, such that Ji(tεφε) ≥ C > 0. Then from the continuity of Ji , we may assume that tε ≥ t1 > 0, where t1 is a positive constant. On the other hand, from the definition of φε and (4.6), for any ε > 0 small enough, we have Ji ( tεφε ) ≤ (S3/2 + C1)t 2 + C2t 4 − S3/2 12 t6, where C1, C2 are positive constants, independent of ε. Thus there exists t2 > 0 such that t1 ≤ tε ≤ t2 for each ε > 0. We set h(t) = t2 2 ∫ |∇φε|2 dx− t6 6 ∫ |φε|6 dx. By a direct calculation, we can show that g attains its maximum at t0 = ( ∫ |∇φε|2 dx∫ |φε|6 dx ) 1 4 . Moreover, by (4.6), using the inequality (a+ b)p ≤ ap + p(a+ b)p−1b, which holds for any p ≥ 1 and a, b ≥ 0, we deduce that max t≥0 h(t) = h(t0) = 1 2 (∫ |∇φε|2 dx∫ |φε|6 dx )2/4 ∫ |∇φε|2 dx− 1 6 ( ∫ |∇φεt|2 dx∫ |φε|6 dx )6/4 ∫ R3 |φε|6 dx = 1 3 ∥∇φε∥32 ∥φε∥36 ≤ 1 3 [S3/2 +O(ε)]3/2 [S3/2 +O(ε3)]1/2 ≤ 1 3 S3/2 +O(ε). EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 19 Then we obtain Ji(tεφε) = t2ε 2 ∫ |∇φε|2 dx+ t2ε 2 ∫ V (x)φ2 ε dx+ t4ε 4 ∫ ϕφεφ 2 ε dx − t6ε 6 ∫ φ6 ε dx− λt4ε p ∫ |φε|p dx ≤ 1 3 S3/2 +O(ε) + C1∥φε∥22 + C2∥φε∥412/5 − C3λ∥φε∥pp. To complete the proof, it remains to show that lim ε→0+ C1∥φε∥22 + C2∥φε∥412/5 − C3λ∥φε∥pp ε = −∞. (4.9) In fact, by (4.6) the following estimate holds as ε→ 0: C1∥φε∥22 + C2∥φε∥412/5 − C3λ∥φε∥pp ≤ C4ε+ C5ε 2 − C6λε (6−q)/2. (4.10) If 4 < q < 6, it follows immediately from (4.9) for any λ > 0. If q = 4, one can chose λ = ε−µ, µ > 0 in the above inequality to obtain (4.9). □ Now, we give the splitting lemma of a (PS)c sequence of J , which plays a crucial role for subsequent discussion. Lemma 4.6 (Splitting Lemma). Let {un} ⊂ E be a bounded (PS)c sequence of J at level m ∈ (0, 13S 2 3 ) and assume that un ⇀ û in E. Then, passing to a subsequence, either un strongly converges to û, or setting k̂ ∈ N ∪ {0}, there are sequences {ui}k̂i=1 ⊂ E and yin ∈ T1Z× T2Z× T3Z ⊂ R3 with 1 ≤ j ≤ k̂ such that (i) (−1)εi−1(yin)1 → +∞ and |(yin)1 − (yjn)1| → +∞ for 1 ≤ i ̸= j ≤ k̂, as n→ +∞, where (yin)γ denotes the γ-th component of yin, 1 ≤ γ ≤ 3. (ii) un → û+ ∑k̂ i=1 u i(· − yin) in E. (iii) J (un) = I(û) + ∑k̂ i=1 Jεi(u i) + on(1). (iv) J ′ εi(u i) = 0 and ui ̸= 0 with 1 ≤ i ≤ k̂. where εi = {1, 2}. Proof. Let us divide the proof in various steps. Step 1: Let z1n = un − û. We have two possibilities: If z1n → 0 in E, then the first alternative follows and the proof is concluded. If z1n ̸→ 0 in E. Let εi = {1, 2} and R3 (−1)εi = { R3 −, εi = 1, R3 +, εi = 2. (4.11) Since z1n ⇀ 0 in E, it follows from Lemmas 2.2 and 2.3 that, as n→ ∞, J (un) = J (z1n) + J (û) + on(1), and for any φ ∈ C∞ 0 (E,R), ε1 = {1, 2} ⟨J ′(un), φ⟩ = ⟨J ′(û), φ⟩+ ⟨J ′(z1n), φ⟩+ on(1) = ⟨J ′(û), φ⟩+ ⟨J ′ ε1(z 1 n), φ⟩+ ∫ R3 (−1)ε1 (−1)ε1(V1 − V2)z 1 nφdx+ on(1) = ⟨J ′(û), φ⟩+ ⟨J ′ ε1(z 1 n), φ⟩+ on(1), 20 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 which together with J ′(un) → 0 and J ′(û) = 0 imply that ⟨J ′ ε1(z 1 n), φ⟩ → 0, as n→ ∞. (4.12) In addition, since ⟨J ′(un), φ⟩ = ⟨J ′(û), φ⟩ + ⟨J ′(z1n), φ⟩ + on(1), which together with J ′(un) → 0 and J ′(û) = 0 implies that ⟨J ′(z1n), φ⟩ → 0, as n→ ∞. Setting φ = z1n, we obtain ⟨J ′(z1n), z 1 n⟩ → 0, i.e., ∥z1n∥2 + ∫ ϕz1 n (z1n) 2 dx− ∫ |z1n|6 dx− λ ∫ |z1n|p dx→ 0. (4.13) Let δ1 := lim sup n→∞ sup y∈R3 ∫ B1(y) |z1n|2 dx. We claim that δ1 ̸= 0. If not, by Lion’s lemma, z1n → 0 in Lr(R3) for r ∈ (2, 6). By Hölder’s inequality, we obtain that∫ ϕz1 n (z1n) 2 dx ≤ (∫ |ϕz1 n |6 dx )1/6(∫ |z1n|12/5 dx )5/6 ≤ CS∥z1n∥212/5 → 0. It follows from (4.13) that ∥z1n∥2 = ∫ |z1n|6 dx + on(1). Asume that ∥z1n∥2 → η1, so ∫ |z1n|6 dx → η1, as n → ∞. By Sobolev embdding, we obtain ∫ |z1n|6 dx ≤ S−1∥z1n∥6, which implies that η1 ≤ S−3η31 . Hence, η1 = 0 or η1 ≥ S3/2. If η1 = 0, then z1n → 0 in E. If η1 ≤ S−3η31 , then m = lim n→∞ J (un) = lim n→∞ J (z1n) + J (û) ≥ lim n→∞ J (z1n) = 1 2 ∥z1n∥2 + 1 4 ∫ ϕz1 n (z1n) 2 dx− 1 6 ∫ |z1n|6 dx− λ p ∫ |z1n|p dx = 1 2 η1 − 1 6 η1 = 1 3 η1 ≥ 1 3 S3/2, which contradicts m ∈ (0, 13S 3/2). Then δ1 ̸= 0. Hence, there exists y1n ∈ T1Z × T2Z× T3Z ⊂ R3 such that ∫ B1(y1 n) |z1n|2dx ≥ δ1 2 . (4.14) Let ξ1n := (0, (y1n)2, (y 1 n)3), σ 1 n := ((y1n)1, 0, 0), and w 1 n = z1n(·+ ξ1n). Clearly, ∥w1 n∥ = ∥z1n∥ and w1 n ⇀ 0 in E but not strongly. Therefore, by (4.14), we obtain∫ B1(σ1 n) |w1 n|2dx ≥ δ1 2 . (4.15) It is easy to check that |(y1n)1| = |σ1 n| → +∞, that is, (−1)ε1−1(y1n)1 → +∞. (4.16) Considering the sequence {w1 n(·+ σ1 n)}, which is bounded in Eε1 , then there exists u1 ∈ Eε1 satisfying w1 n(·+ σ1 n)⇀ u1 in Eε1 , w1 n(·+ σ1 n) → u1 in Lr loc (R3), EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 21 w1 n(x+ σ1 n) → u1(x) a.e. on R3. From (4.15), we obtain u1 ̸= 0. From (4.11), for any φ ∈ C∞ 0 (E,R), we obtain ⟨J ′ ε1(w 1 n(·+ σ1 n)), φ⟩ = ⟨J ′ ε1(z 1 n(·+ y1n)), φ⟩ = ⟨J ′ ε1(z 1 n), φ(· − y1n)⟩ → 0. Hence, we have J ′ ε1(w 1 n(·+ σ1 n)) → 0 since C∞ 0 (E,R) is dense in E, and then J ′ ε1(u 1) = 0. (4.17) Step 2: Let z2n = z1n − u1(x − y1n). Then, z2n ⇀ 0 in E due to the norm of E is equivalent to the norm of Eε1 . In addition, from Lemmas 2.2 and 2.3, by the simple calculation, as n→ ∞, J (un) = 1 2 ∥un∥2 + 1 4 ∫ ϕunu 2 n dx− 1 6 ∥un∥66 − λ p ∥un∥pp = J (z1n) + J (û) + on(1) = J (z2n) + J (u1) + J (û) + on(1) = J (z2n) + Jε1(u 1) + ∫ R3 (−1)ε1 (−1)ε1(V1 − V2)u 1 dx+ J (ū) + on(1) = J (z2n) + Jε1(u 1) + J (ū) + on(1). (4.18) We have two possibilities: If z2n → 0 in E, i.e., z1n − u1(x− y1n) = un − û− u1(x− y1n) → 0, i.e., un → û+ u1(x− y1n). (4.19) From (4.18), we obtain J (un) = J (û) + Jε1(u 1) + on(1). (4.20) Then the Lemma is proved for k = 1. It follows from (4.16), (4.17), (4.19), and (4.20). If z2n ̸→ 0 in E. From Lemmas 2.2 and 2.3, we obtain for any φ ∈ C∞ 0 (E,R), as n→ ∞, ⟨J ′(z1n), φ⟩ = ⟨J ′(z2n), φ⟩+ ⟨J ′(u1), φ(x+ y1n)⟩+ on(1) = ⟨J ′(z2n), φ⟩+ ⟨J ′ ε1(u 1), φ(x+ y1n)⟩ + ∫ R3 (−1)ε1 (−1)ε1(V1 − V2)u 1φ(x+ y1n)dx+ on(1) = ⟨J ′(z2n), φ⟩+ ⟨J ′ ε1(u 1), φ(x+ y1n)⟩+ on(1). (4.21) It follows from ⟨J ′(z1n), φ⟩ → 0 and J ′ ε1(u 1) = 0 that we obtain ⟨J ′(z2n), φ⟩ → 0. Setting φ = z2n, we obtain ⟨J ′(z2n), z 2 n⟩ → 0, i.e., ∥z2n∥2 + ∫ ϕz2 n (z2n) 2 dx− ∫ |z1n|6 dx− λ ∫ |z2n|p dx→ 0. (4.22) Let δ2 := lim sup n→∞ sup y∈R3 ∫ B1(y) |z2n|2dx. Simlar to δ1 ̸= 0, we obtain δ2 ̸= 0. Then, there exists y2n ∈ T1Z× T2Z× T3Z ⊂ R3 such that ∫ B1(y2 n) |z2n|2dx ≥ δ2 2 . 22 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 Let ξ2n := (0, (y2n)2, (y 2 n)3), σ 2 n := ((y2n)1, 0, 0), and w 2 n = z2n(x+ξ 2 n). Clearly, ∥w2 n∥ = ∥z2n∥ and w2 n ⇀ 0 in E. Therefore,∫ B1(σ2 n) |w2 n|2dx ≥ δ2 2 . (4.23) It is easy to check that as n→ ∞, |(y2n)1| = |σ2 n| → +∞, that is (−1)ε2−1(y2n)1 → +∞. (4.24) Then w2 n(·+ σ2 n) ̸⇀ 0 in E. In addition, we claim that |(y2n)1 − (y1n)1| → +∞. (4.25) To see this, first observe that w2 n(·+ σ2 n + (y2n)1 − (y1n)1) = z2n(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n) = z1n(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n)− u1(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n − y1n) = w1 n(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n − ξ1n)− u1(·+ σ2 n + (y2n)1 − (y1n)1 + ξ2n − y1n) = w1 n(·+ σ2 n + y2n − y1n)− u1(·+ σ2 n + y2n − y1n − σ1 n). From Lemma 3.4 (ii), since w2 n(·+ σ2 n) ̸⇀ 0 in E, if it were |(y2n)1 − (y1n)1| ̸→ +∞, we obtain w2 n(·+ σ2 n + (y2n)1 − (y1n)1) ̸⇀ 0. On the other hand, since w1 n(·+ σ1 n)⇀ u1, we obtain w1 n(·+ σ2 n + y2n − y1n)− u1(·+ σ2 n + y2n − y1n − σ1 n)⇀ 0, which is a contradiction. So we obtain (4.25). Moreover, for any φ ∈ C∞ 0 (E,R), we obtain as n→ ∞, ⟨J ′ ε2(z 1 n), φ⟩ = ⟨J ′ ε2(z 2 n), φ⟩+ ⟨J ′ ε1(u 1), φ(x+ y1n)⟩+ ∫ (Vε2 − Vε1)u 1φ(x+ y1n)dx+ on(1) = ⟨J ′ ε2(z 2 n), φ⟩+ ⟨J ′ ε1(u 1), φ(x+ y1n)⟩+ on(1). By ⟨J ′ ε2(z 1 n), φ⟩ → 0 and J ′ ε1(u 1) = 0, we obtain J ′ ε2(z 2 n) → 0 since C∞ 0 (E,R) is dense in E. Considering the sequence { w2 n(·+ σ2 n) } , which is bounded in Eε2 , there exists u2 ∈ Eε2 satisfying w2 n(·+ σ2 n)⇀ u2 in Eε2 , w2 n(·+ σ2 n) → u2 in Lr loc (R3), w2 n(x+ σ2 n) → u2(x) a.e. on R3. We can see that u2 ̸= 0. For any φ ∈ C∞ 0 (E,R), we obtain ⟨J ′ ε2(u 2), φ⟩ = ⟨J ′ ε2(w 2 n(·+σ2 n)), φ⟩ = ⟨J ′ ε2(z 2 n(·+y2n)), φ⟩ = ⟨J ′ ε2(z 2 n), φ(·−y2n)⟩ → 0. Hence, since C∞ 0 (E,R) is dense in E, we have I ′ ε2(u 2) = 0. (4.26) EJDE-2024/43 SCHRÖDINGER-BOPP-PODOLSKY SYSTEMS 23 Step 3: Let z3n = z2n − u2(x− y2n). From (4.18), we have J (un) = J (z2n) + Jε1(u 1) + J (ū) + on(1) = J (z3n) + Jε2(u 2) + Iε1(u1) + J (ū) + on(1). (4.27) We have two possibilities: If z3n → 0 in E, i.e., z2n−u2(x− y2n) = un− ū−u1(x− y1n)−u2(x− y2n) → 0, i.e., un → ū+ u1(x− y1n) + u2(x− yn). (4.28) From (4.27), we obtain J (un) = J (ū) + Jε1(u 1) + Jε2(u 2) + on(1). (4.29) Then the Lemma is proved for k = 2 follows from (4.24), (4.25), (4.28), (4.29) and (4.26). If z2n ̸→ 0 in E, we just repeat the argument. Step k̂: By J (un) = J (ū)+ ∑k̂ i=1 Jεi(u i)+on(1), since Jεi(u i) ≥ cεi ≥ min{c1, c2} and J (un) is bound, the iteration must stop at some finite index k̂. The proof is complete. □ Proof of Theorem 1.3. In view of Lemma 4.4, there exists a bounded (PS)c se- quence {uk} ⊂ M such that J (uk) → m and J ′(uk) → 0 as k → +∞. Since {uk} is bounded in E, going to a subsequence if necessary, still denoted by {uk}, we can suppose that there exists û ∈ E such that uk ⇀ û in E. Withm < min{m1,m2}, by Lemma 4.6, if uk ̸→ u, we can show that k ≥ 1 and nontrivial solutions u1, u2, . . . , uj of Jεj with εj = {1, 2} satisfy m = lim k→+∞ J (uk) = I(û) + k̂∑ j=1 Jεj (u j) ≥ k̂min{m1,m2} ≥ min{m1,m2}, which contradicts m < min{m1,m2}. Thus, uk → û, and then J (û) = m and J ′(û) = 0. Obviously, û ̸= 0. Therefore, û is a ground state solution of (1.14). Considering û0 = |û|, it is easy to check that J (û0) = J (û) = m and û0 ∈ M . From standard arguments, we infer that J ′(û0) = 0. Thus, û0 is a non-negative solution of system (1.14). Furthermore, the strong maximum principle implies that û0 > 0 in R3, and thus, û0 is a positive ground state solution of system (1.14). □ Proof of Theorem 1.4. We just study the case of m1 ≤ m2 since the case m2 ≤ m1 is analogous. Let v1 ∈ M1 ⊂ E be a positive ground state for the purely periodic problem for (1.16) with i = 1. We can see from Lemma 2.2 that there exists ŝ > 0 satisfying ŝv1 ∈ M. Then, from the assumption of Theorem 1.4 and Corollary 4.3, we have m ≤ J (ŝv1) = ŝ2 2 ∥v1∥2 + ŝ4 4 ∫ ϕv1 |v1|2 dx− ŝ6 6 ∥v1∥pp − λŝp p ∥v1∥pp = ŝ2 2 ∥v1∥2E1 + ŝ4 4 ∫ ϕv1 |v1|2 dx− ŝ6 6 ∥v1∥pp − λŝp p ∥v1∥pp + ŝ2 2 ∫ R3 − (V2 − V1)v 2 1 dx = J1(ŝv1) + ŝ2 2 ∫ R3 − (V2 − V1)v 2 1 dx 24 Q. JIANG, L. LI, S. CHEN, G. SICILIANO EJDE-2024/43 < J1(v1) = m1, which implies that m < m1. Thus, m < min{m1,m2}. □ Acknowledgements. 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School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China Email address: lilin420@gmail.com Shangjie Chen School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China Email address: 11183356@qq.com Gaetano Siciliano Dipartimento di Matematica, Univeristà degli Studi di Bari, via E. Orabona 4, 70215 Bari, Italy Email address: gaetano.siciliano@uniba.it 1. Introduction 2. Preliminary results 3. Subcritical case 4. Critical case Acknowledgements References