The existence of nontrivial solution of a class of Schrödinger–Bopp–Podolsky system with critical growth

where λ > 0 is a parameter. Systems such as (1.1) have been introduced in [1] as a model describing solitary waves for nonlinear stationary equations of Schrödinger type interacting with an electrostatic field in the Bopp–Podolsky electromagnetic theory and are usually known as Schrödinger–Bopp–Podolsky systems. We refer to [2–7] for a more detailed description of the physical aspects of this problem. In this paper, we suppose that V , f satisfy the following assumptions: (V1) V ∈ C(R3,R), V0 = infx∈R3 V (x) > 0.


Introduction and statement of results
In this paper, we deal with the following Schrödinger-Bopp-Podolsky system with critical growth: where λ > 0 is a parameter. Systems such as (1.1) have been introduced in [1] as a model describing solitary waves for nonlinear stationary equations of Schrödinger type interacting with an electrostatic field in the Bopp-Podolsky electromagnetic theory and are usually known as Schrödinger-Bopp-Podolsky systems. We refer to [2][3][4][5][6][7] for a more detailed description of the physical aspects of this problem. In this paper, we suppose that V , f satisfy the following assumptions: (V 2 ) For any T > 0, there exists r > 0 such that where meas(A) is the Lebesgue measure of A. (f 1 ) f ∈ C(R) and f (u) = o(u) as u → 0. (f 2 ) f (u)/u → +∞ as |u| → ∞. The solution to (1.1) is understood in the weak sense, that is, a pair (u, φ) ∈ H 1 (R 3 ) × D is a solution to (1.1) if where D is a function space that will be introduced in Sect. 2. To the best of our knowledge, there are very few papers related to the existence of solutions to problem (1.1). In [1], d' Avenia and Siciliano studied the following Schrödinger-Bopp-Podolsky equation: (1. 2) The authors give existence and nonexistence results, depending on the parameters p and q. Moreover, they also show that in the radial case, the solutions that they find tend to solutions of the classical Schrödinger-Poisson system as a → 0. When a = 0, (1.2) reduces to the following well-known Schrödinger-Poisson equation that has been extensively studied in the past few decades. There have been many existence and nonexistence results in the past decades. For some recent results, we refer the readers to [8][9][10][11][12][13] and the references therein. We now summarize our main results as follows.
Remark 1.1 We note that the usual growth condition and the Ambrosetti-Rabinowitz condition are not needed in our result. Moreover, f is allowed to be sign-changing.
Remark 1.2 A typical example of a function satisfying assumptions (f 1 )-(f 2 ) is given by f (t) = |t| q-2 t, q > 6. Furthermore, our conclusion holds for general supercritical nonlinearity.
The proof will be carried out by variational methods. Since the Sobolev embedding H 1 (R 3 ) → L s (R 3 ) is not compact, the main difficulty is the lack of compactness. Since we do not assume Ambrosetti-Rabinowitz or growth conditions on f , we first make a suitable modification on f , solve the modified problem, and then check that, for small enough λ, the solutions of the modified problem are also the solutions of the original problem. We note that even for the modified problem it is not easy to obtain compactness in view of the critical growth of the nonlinearity. To overcome the loss of compactness for the energy functional, we shall verify that the Palais-Smale condition is regained when the energy functional is below a suitable level.
The rest of this paper is organized as follows. In Sect. 2, we state some preliminary notations, modify the original problem, and prove the existence result of the modified problem. In Sect. 3, we prove Theorem 1.1.

Preliminaries and the modified problem
In this paper, we use the following notation: • H 1 (R 3 ) is the usual Sobolev space with an inner product and norm given by • L p (R 3 ), 1 ≤ p ≤ +∞, denotes a Lebesgue space, and the norm in • C, C i denote (possible different) any positive constant.
In this section, we summarize some fundamental properties of the operator -+ 2 and functional space D. The D is defined by the completion of C ∞ 0 (R 3 ) equipped with the norm · D induced by the scalar product For more details, we refer the reader to [1].
It is easy to show that D is a Hilbert space continuously embedded into D 1,2 (R 3 ) and consequently in L ∞ (R 3 ), see [1].
For every fixed u ∈ H 1 (R 3 ), the Riesz theorem implies that there exists a unique solution φ u ∈ D such that φ + 2 φ = u 2 .
In order to write explicitly this solution, we consider The function φ u possesses the following properties (see [1]).

Lemma 2.2 For every u
Substituting (2.1) into (1.1), we obtain Then we define a smooth functional Φ : In fact, functional Φ possesses the following useful BL-splitting properties, similar to the Brézis-Lieb lemma [14].

Consequently, by (2.3) and Lemma 2.2 (ii)-(iii), we obtain that
The proof is complete.
We shall search critical points for the functional It is well defined on the Hilbert space and has the inner product and norm It is well known under assumptions (V 1 ) and (V 2 ) that we have the following compactness lemma see [15] or [16].

Lemma 2.4
Suppose that assumptions (V 1 ) and (V 2 ) are satisfied. Then the embedding from X into L s (R 3 ) is compact for s ∈ [2, 6).
Since f is continuous, we have I λ ∈ C 1 (X, R) and t = +∞, we can introduce a truncated function. Let T > 0 be large enough such that f (T) > 0 according to (f 2 ). We set . Based on assumptions (f 1 ) and (f 2 ) it is easy to show that g T (t) is a continuous function and satisfies the following properties: We shall search critical points for the functional as solutions to (2.4). Since g T is continuous, we have I λ,T ∈ C 1 (X, R) and, for any u, v ∈ X, The next lemma shows that the functional I λ,T (u) satisfies the mountain pass geometry [14].

Lemma 2.5
The functional I λ,T (u) satisfies the following conditions: (ii) there exists e ∈ X such that e > ρ and I λ,T (e) < 0.
Proof For any u ∈ X\{0} and > 0 small, it follows from (g 1 ) and (g 3 ) that g T (t) ≤ |t| + C |t| 5 and G T (t) ≤ 2 |t| 2 + C 6 |t| 6 . Thus by Lemma 2.2 (i) and the Sobolev embedding X → L s (R 3 ) for s ∈ [2,6]. Since is arbitrarily small, there exist ρ > 0 and α > 0 such that I λ,T (u) ≥ α > 0 for u = ρ. Let us check (ii). From (g 2 ), for any M > 0, there exists r M > 0 such that Together with (g 1 ) and (g 3 ), this implies that, for any M > 0, there exists a constant C M > 0 such that Then, for each u ∈ X \ {0} and t > 0, we obtain that The step is proved by taking e = t 0 u with t 0 > 0 large enough. Now, in view of Lemma 2.5, we can apply a version of the mountain pass theorem without the (PS) condition to obtain a sequence {u n } such that As in [14], we define

Lemma 2.6 Every sequence satisfying (2.7) is bounded in X.
Proof For every c ∈ R, let {u n } ⊂ X be a (PS) c sequence satisfying (2.7). Then, by (g 4 ), we deduce that where u + n = max{u n (x), 0}, un = min{u n (x), 0}, u n (x) = u + n + un . We argue by contradiction that u n → +∞ as n → ∞. Let v n = u n u n , n ≥ 1.
We conclude that v + = 0. Then u + n = v + n u n → +∞. By (2.5) and (2.7), we obtain (2.10) Taking the limit and using Lemma 2.2 (ii), (g 2 ), and (2.9), we obtain 0 ≤ -∞, yielding a contradiction. Therefore, {u n } is bounded in X. Now, we denote by S the best constant of the Sobolev embedding H 1 (R 3 ) → L 6 (R 3 ), i.e., As we will show in the following result, the modified functional satisfies the local compactness condition. Proof Let {u n } be a (PS) c λ,T sequence satisfying (2.7). By Lemma 2.6, {u n } is bounded in X. Up to a subsequence, we may assume that u n → u, a.e. in R 3 , u n u, weakly in X, u n → u, strongly in L s R 3 , 2 ≤ s < 6. (2.11) Since φ : L 12/5 (R 3 ) → D is continuous, from (2.11) we obtain that (2.12) Using (2.11) and [14, Theorem A.1 ], for any ϕ ∈ C ∞ 0 (R 3 ) ⊂ X, we can obtain that (2.13) From (2.11)-(2.12), the Hölder inequality, and the Sobolev embedding, we obtain (2.14) By (2.13)-(2.14), the density of C ∞ 0 (R 3 ) in X, and (2.7), we can conclude that I λ,T (u n ) → I λ,T (u) = 0. Let w n = u nu, as n → ∞. It follows from Lemma 2.3 and the Brezis-Lieb lemma that and I λ,T (w n ) → 0 in X -1 . We recall that the continuous embedding X → L s (R 3 ) is compact for 2 ≤ s < 6. Hence, up to a subsequence, w n → 0 in L s (R 3 ), and Since w n ⊂ X is bounded, we may assume that as n → ∞ up to a subsequence. Suppose by contradiction that b > 0. By the Sobolev inequality, we have and, therefore, b ≥ S 3 2 . Thus which contradicts our assumption. Therefore, b = 0 and the proof is complete.
To obtain the existence result for problem (2.4) by Lemma 2.7, we need to show that the mountain pass value c λ,T < 1 3 S
Proof Since the functional I λ,T contains the mountain pass geometry and satisfies the (PS) c condition, the mountain pass theorem [14] implies that there exists a critical point u λ ∈ X. Moreover, I λ,T (u λ ) = c λ,T ≥ α > 0 = I(0), so that u λ is a nontrivial solution.

Proof of Theorem 1.1
In this section, we prove our main result. Our approach is based on showing that the solution obtained in Theorem 2.9 satisfies the estimate |u λ | ∞ ≤ T. This implies that u λ is indeed the solution to the original problem (1.1). The following lemma plays a fundamental role in the study of the existence of the nontrivial solution to problem (1.1), and its proof involves some arguments explored in [17,18] and involves the use of the Nash-Moser method [19].