Ground state solutions of the non-autonomous Schrödinger–Bopp–Podolsky system

In this paper, we consider the following non-autonomous Schrödinger–Bopp–Podolsky system -Δu+V(x)u+q2ϕu=f(u)-Δϕ+a2Δ2ϕ=4πu2inR3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + V(x) u + q^2\phi u = f(u)\\ -\Delta \phi + a^2 \Delta ^2 \phi = 4\pi u^2 \end{array}\right. } \hbox { in }{\mathbb {R}}^3. \end{aligned}$$\end{document}By using some original analytic techniques and new estimates of the ground state energy, we prove that this system admits a ground state solution under mild assumptions on V and f. In the final part of this paper, we give a min-max characterization of the ground state energy.

This nonlinear system appears when we couple a Schrödinger field ψ = ψ(t, x) with its electromagnetic field in the Bopp-Podolsky electromagnetic theory, and, in particular, in the electrostatic case for standing waves ψ(t, x) = e iωt u(x).
System (1.1) has a strong physical meaning especially in the Bopp-Podolsky theory, developed independently by Bopp [3] and Podolsky [24]. The Bopp-Podolsky theory is a second order gauge theory for the electromagnetic field. As the Mie theory [22] and its generalizations given by Born and Infeld [4][5][6][7], it was introduced to solve the "infinity problem", which appears in the classical Maxwell theory. In fact, by the well-known Gauss law (or Poisson equation), the electrostatic potential φ for a given charge distribution whose density is ρ satisfies the equation If ρ = 4πδ x 0 , with x 0 ∈ R 3 , the fundamental solution of (1.2) is G(x − x 0 ), where G(x) = 1 |x| , and the electrostatic energy is Thus, Eq. (1.2) is replaced by − div ∇φ in the Born-Infeld theory and by − φ + a 2 2 φ = ρ in R 3 in the Bopp-Podolsky theory. In both cases, if ρ = 4πδ x 0 , we are able to write explicitly the solutions of the respective equations and to see that their energy is finite.
In particular, when we consider the differential operator − + a 2 2 , we have that is the fundamental solution of the equation Then K has no singularity in x 0 since it satisfies and its energy is Moreover, the Bopp-Podolsky theory may be interpreted as an effective theory for short distances (see [20]), while for large distances it is experimentally indistinguishable from the Maxwell theory. Thus, the Bopp-Podolsky parameter a > 0, which has dimension of the inverse of mass, can be interpreted as a cut-off distance or can be linked to an effective radius for the electron. For more physical details we refer the reader to the recent papers [1,2,9,10,16,17] and to references therein.
The differential operator − + 2 appears in various different interesting mathematical and physical situations; see [19] and the references therein.
Before stating our results, few preliminaries are in order. We introduce here the space D as the completion of C ∞ c (R 3 ) with respect to the norm ∇φ 2 2 + a 2 φ 2 2 ; see Sect. 2 for more properties on this space.
For fixed a > 0 and q = 0, we say that a pair (u, φ) ∈ H 1 (R 3 ) × D is a solution of problem (1.1) if We say that a solution (u, φ) is nontrivial whenever u ≡ 0; a solution is called a ground state solution if its energy is minimal among all nontrivial solutions. As described in Sect. 2, to solve problem (1.1) is equivalent to solving 3) whose solutions correspond to critical points of the energy functional defined in H 1 (R 3 ) by (1.4) where F(u) = u 0 f (t)dt. A solution is called a ground state solution if its energy is minimal among all nontrivial solutions.
In this paper, we also consider the following "limit" system with a general nonlinearity f To the best of our knowledge, there is no result on the existence of ground state solutions for systems (1.1) and (1.5). Inspired by [11,12,14,25], we will seek a ground state solution of Nehari-Pohozaev type for systems (1.1) and (1.5).
Our first result is as follows. For the constant potential case, we replace the monotonicity condition (F4) with the super-quadratic condition which is easier to verify: (F5) f (t)t ≥ 3F(t) for all t ∈ R, and there exist κ > 3/2 and r 0 , C 0 > 0 such that Our second result is as follows. Finally, we give the min-max property of the ground state energy of I. To this end, we introduce the following monotonicity condition.
We define the Nehari-Pohozaev manifold as follows: where P(u) is the Pohozaev functional of (1.3) defined by is a critical point of I, then u satisfies P(u) = 0; see [18, A.14] for more details. Then every nontrivial solution of (1.1) is contained in M. In this direction, we have the following theorem.
To state the following result, we define the energy functional in H 1 (R 3 ) by (1.8) and the Nehari-Pohozaev manifold by where P ∞ (u) is the Pohozaev functional defined by We have the following corollary.  (1.10) Next, we show that the sequence {u n } is bounded in H 1 (R 3 ). Due to lack of global compactness and adequate information on I (u n ) and in order to avoid relying the radial compactness, we establish a crucial inequality related to I(u), I(u t ) and J (u) (Lemma 3.4), which plays a crucial role in our arguments, see Lemmas 3.8, 3.9, 3.13, 3.14 and 4.5 . With the help of this inequality, we then can recover the compactness for the minimizing sequence {u n } and show that {u n } converges weakly to someū ∈ H 1 (R 3 )\{0} and I(ū) = inf M I by using Lions' concentration-compactness, the "least energy squeeze approach" and some subtle analysis. Finally, we take advantage of a quantitative deformation lemma and the intermediate value theorem to show thatū is a critical point of I, as the Lagrange multiplier theorem does not work, because M is not a C 1 -manifold, .
To prove Theorem 1.1, we use the monotonicity technique explored by Jeanjean [21] to parameterize the nonlinearity f . In such a way, we build a parametrization of the energy functional associated to (1.1) and give some energy relations of problems (1.1) and (1.5) which play a key role in getting the critical point of (1.1), see Lemma 4.5.
Moreover, in order to show that a critical point associated to the parametrization functional is indeed a solution to the original problem, we also need give a delicate estimation for the parametrization problem. Finally, we study the constant potential case by using weaker conditions.
Throughout the paper we make use of the following notations: • Under (V1), H 1 (R 3 ) denotes the Sobolev space equipped with the inner product and norm positive constants possibly different in different places.

Variational setting
We start with some preliminary basic results. Let us consider the nonlinear Schrödinger Lagrangian density where ψ : R × R 3 → C, , m > 0, and let (φ, A) be the gauge potential of the electromagnetic field (E, H), namely φ : R 3 → R and A : R 3 → R 3 satisfy The coupling of the field ψ with the electromagnetic field (E, H) through the minimal coupling rule, namely the study of the interaction between ψ and its own electromagnetic field, can be obtained by replacing in L Sc the derivatives ∂ t and ∇ respectively with the covariant ones q being a coupling constant. This leads to consider Now, to get the total Lagrangian density, we have to add to L CSc the Lagrangian density of the electromagnetic field.
The Bopp-Podolsky Lagrangian density (see [24,Formula (3.9)]) is Thus, the total action is Then D is a Hilbert space continuously embedded into D 1,2 (R 3 ) and consequently in L 6 (R 3 ).
We notice the following auxiliary properties; see Lemmas 3.1 and 3.2 in [18].
The next property gives a useful characterization of the space D.
For every fixed u ∈ H 1 (R 3 ), the Riesz representation theorem implies that there is a unique solution φ u ∈ D of the second equation in (1.1). To write explicitly such a solution (see also [24,Formula (2.6)]), we consider We have the following fundamental properties.

Lemma 2.3 [18, Lemma 3.3] For all y ∈ R 3 , K(· − y) solves in the sense of distributions
Moreover, In both cases, K * g solves in the sense of distributions, and we have the following distributional derivatives: Actually the following useful properties hold.

Lemma 2.4 [18, Lemma 3.4]
For every u ∈ H 1 (R 3 ) we have: Under hypotheses (V1), (F1) and (F2), the energy functional defined in is continuously differentiable and its critical points correspond to the weak solutions (2.4) In order to avoid the difficulty generated by the strongly indefiniteness of the functional S, we apply a reduction procedure.
an application of the implicit function theorem gives Jointly with (2.3) and (2.4), the functional I(u) := S(u, φ u ) has the reduced form Moreover, the following statements are equivalent: is a critical point of I, then the pair (u, φ u ) is a solution of (1.1). For the sake of simplicity, in many cases we just say

Proof of Theorem 1.3
In this section, we give the proof of Theorem 1.3.
By a simple calculation, we have the following two lemmas.
Note that if t → 0 in (3.4) and (3.5), then and Proof Arguing by contradiction, we assume that there exist a sequence {x n } ⊂ R 3 and δ > 0 such that Then it follows from (V1), (3.9) and (3.11) that which is an obvious contradiction.
Remark that (3.18) with t → 0 gives For the limiting problem, corresponding to (2.5) and (3.16), we define the following functionals in H 1 (R 3 ): and From Lemma 3.4, we deduce the following two properties.
(3.23) By using (3.5) instead of (3.4), as in the proof of Lemma 3.4, we have the following lemma.
This shows that { u n 2 } is bounded. Now we assert that { ∇u n 2 } is also bounded. Arguing by contradiction, suppose that ∇u n 2 → ∞. From (F1), (F2) and the Sobolev inequality, there exists C 2 > 0 such that This contradiction shows that { ∇u n 2 } is also bounded and the assertion holds. Hence {u n } is bounded in H 1 (R 3 ). Thus, there existsū ∈ H 1 (R 3 ) such that, passing to a subsequence, u n ū in H 1 (R 3 ), u n →ū in L s loc (R 3 ) for all 1 ≤ s < 6 and u n →ū a.e. in R 3 . There are two possible cases: i)ū = 0 and ii)ū = 0.

(3.43)
Let w n =û n −û. Then (3.43) and Lemma 3.12 yield We define the functional ∞ : which contradicts the fact that ∞ (û) > 0. Hence, J ∞ (û) ≤ 0 and the claim holds. In view of Lemma 3.8, there exists t ∞ > 0 such that t 2 ∞û t ∞ ∈ M ∞ . Now which implies again the validity of (3.47) also in this case. In view of Lemma 3.8, there existst > 0 such thatt 2ût ∈ M. Moreover, it follows from (V1), (2.5), (3.21), (3.47) and Corollary 3.5 that This shows that m is achieved att 2ût ∈ M. Case ii)ū = 0. We define the functional : (3.48) In this case, similarly to the proof of (3.47), by using I, J and instead of I ∞ , J ∞ and ∞ , we deduce that I(ū) = m and J (ū) = 0. Proof Assume that I (ū) = 0. Then there exist δ > 0 and ρ > 0 such that It is easy to check that Then there exists δ 1 > 0 such that Using (V1), (V3) and (F1)-(F3), it is easy to prove that there exist T 1 ∈ (0, 1) and In view of Lemma 3.4, we have The rest of the proof is similar to that of [11,Lemma 2.14]. For the sake of completeness, we give the details. Let Note that Corollary 3.5 implies that I t 2ū t ≤ I(ū) = m for all t > 0. Then (3.49) and ii) give On the other hand, (3.51) and iii) yield where Define the function 0 (t) := J η 1, t 2ū t for all t > 0. It follows from (3.51) and i) that η(1, t 2ū t ) = t 2ū t for t = T 1 and t = T 2 , which, together with (3.50), implies Since 0 (t) is continuous on (0, ∞), then we have that η 1, t 2ū t ∩ M = ∅ for some t 0 ∈ [T 1 , T 2 ], contradicting the definition of m.
Proof of Theorem 1. 4 In view of Lemmas 3.13 and 3.14, there existsū ∈ M such that This shows thatū is a ground state solution of (1.1) such that I(ū) = m = inf M I.

Remark 3.15
As in the proof of Theorem 1.4, by replacing Lemma 3.4 with Lemma 3.7, we then obtain Corollary 1.6.

Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1. Without loss of generality, we consider that V (x) ≡ V ∞ .
Proposition 4.1 [21] Let X be a Banach space and let J ⊂ R + be an interval, and be a family of C 1 -functionals on X such that

maps every bounded set of X into a set of R bounded below;
(iii) there are two points v 1 , v 2 in X such that Then, for almost every λ ∈ J , there exists a sequence {u n (λ)} such that (ii) λ (u n (λ)) →c λ ; (iii) λ (u n (λ)) → 0 in X * , where X * is the dual of X .
Similarly, for all λ ∈ [1/2, 1], if u is a critical point of I ∞ λ , then u satisfies the following Pohozaev-type identity: We also let By Lemma 3.7, we have the following lemma.

Conflict of interest
The authors declare that they have no conflict of interest.
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