Infinitely many solutions for a gauged nonlinear Schrödinger equation with a perturbation

*This work is supported by the China Postdoctoral Science Foundation (grant No. 2019M652348), Natural Science Foundation of Chongqing (grant No. cstc2020jcyj-msxmX0123), Technology Research Foundation of Chongqing Educational Committee (grant Nos. KJQN201900539 and KJQN202000528), Key Research Funds for the Universities of Henan Province (19A110018, 20B110006), Fundamental Research Funds for the Universities of Henan Province (NSFRF180320), Henan Polytechnic University Doctor Fund (B2016-58). 1Corresponding author.


Introduction
In this paper, we study the existence of infinitely many high energy solutions for the following gauged nonlinear Schrödinger equation with a perturbation in R 2 : We first list our assumptions for our problem (1): (V1) V ∈ C(R 2 , R), and inf x∈R 2 V (x) V 0 > 0, where V 0 is a positive constant. (V2) There exists b > 0 such that meas{x ∈ R 2 : V (x) b} is finite; here meas denotes the Lebesgue measure.
There also are some papers in the literature, which consider perturbation terms; see [15,17,22,23,27] and the references therein. In [15,17], the authors used the famous Ambrosetti-Rabinowitz conditions to study the existence of solutions for the following fractional equations: and where (−∆) s p is the fractional p-Laplacian operator, and (I − ∆) s is the fractional Bessel operator. Moreover, [17] also considered the effect of the parameter λ, µ on the existence of solutions for their problem.
Motivated by the aforementioned works, in this paper, we study the existence of infinitely many high energy solutions under some appropriate conditions, which are weaker than the Ambrosetti-Rabinowitz conditions, and also consider the effect of the parameters and the perturbation terms on the existence of solutions. Now, we state our main result: Theorem 1. Suppose that (V1), (V2), (H1)-(H5), and (g) hold. Then for any µ > 0, there exists Λ > 0 such that system (1) possesses infinitely many high energy solutions when λ Λ.

Preliminaries
Note the parameter λ, and we can consider the work space Then E is a Hilbert space with the inner product and norm Moreover, by [24] we have that the embedding E → L r (R 2 ) is continuous for r ∈ [2, +∞) and E → L r (R 2 ) is compact for r ∈ (2, +∞), i.e., there are constants γ r > 0 such that u r γ r u for 2 r < ∞, where · r is the norm in the usual Lebesgue space L r (R 2 ).
In what follows, we present the energy functional I : E → R for problem (1) defined as Note (3) and (g). We obtain that I is of class C 1 and its derivative is Lemma 1. (See [1,13,14,29].) Suppose that {u n } converges weakly to a function u in E as n → ∞. Then In order to obtain our main result, we need to introduce the Fountain theorem under the Cerami condition (C). [16].) Assume that X is a Banach space. We say that J satisfies the Cerami condition if (C) J ∈ C 1 (X, R), and for all c ∈ R, Then J has a sequence of critical points u n such that J(u n ) → +∞ as n → ∞.
3 Proof of Theorem 1 Lemma 3. Let sequence {u n } converge weakly to a function u in E, u n (x) → u(x) a.e. in R 2 as n → ∞. Then In particular, if and I (u n − u), ϕ = o(1) ∀ϕ ∈ E as n → ∞.
Proof. From the compactness of E → L r (R 2 ), for r ∈ (2, +∞), we have u n u weakly in E, u n → u strongly in L p (R 2 ) for p ∈ (2, +∞), Let w n = u n − u. Then we have w n 0 weakly in E, w n → 0 strongly in L p (R 2 ) for p ∈ (2, +∞), w n → 0 for a.e. x ∈ R 2 .
Since u n u in E, we have (u n − u, u) → 0 as n → ∞, which implies u n 2 = (w n + u, w n + u) = w n 2 + u 2 + o(1) as n → ∞.
Note Lemma 1(v), and we have B(u n − u) C 0 u n − u 4 4 u n − u 2 2 → 0 as n → ∞. Consequently, to obtain (4), by Lemma 1(i) we only need to check that and Note the definition of (·, ·), for all n ∈ N, we have (u n , ϕ) = (u n − u, ϕ) + (u, ϕ). Moreover, since w n 0 in E and by Lemma 1(iii), to prove (5), it suffices to show that and We first prove that (9) and (11). Using the inequality from page 13 in [17] and the Hölder inequality, for qq /(q − 1) > 2, we have Hence, (9) holds. From Lemma 1 in [3] there exists C q > 0 such that ||u n | q−2 u n − |u| q−2 u| C q |u n − u| q−1 . Therefore, from (g) and the Hölder inequality we only need to prove: → 0 as n → ∞.  (10). In what follows, we prove (8). Using the ideas in [17,22,23], we have Hence, from (2) we obtain Therefore, together with (3), using the Young inequality with ε (for all ε > 0), we obtain Consequently, we consider the function f n defined as and by the Lebesgue dominated convergence theorem we have Note that Using (12) shows that (8) holds. Compare (4), (5) with (6), (7). We only need to prove that I (u), ϕ = 0 for all ϕ ∈ E. Note Lemma 1(iii), (10), (11), and (u n − u, ϕ) → 0 as n → ∞. It suffices to check that Note the arbitrariness of ε in (2), and w n → 0 in L p (R 2 ), p > 2. Therefore, from (2) we have This completes the proof. Proof. For all c ∈ R, suppose that there exists {u n } n∈N ⊂ E is bounded and Using I (u), ϕ = 0 for all ϕ ∈ E in Lemma 3 and noting Lemma 1(iv), we have This implies that Recall w n = u n − u. From (6) and (7) we have M for some M > 0.
As V (x) < b on a set of finite measure and w n 0 in E, we have Combining this and the Hölder inequality, recall p = 2τ /(τ − 1) ∈ (2, +∞), fixed ν ∈ (p, +∞), we have From (H1), for all ε > 0, there exists δ = δ(ε) > 0 such that |f (u)| ε|u| for x ∈ R 2 and |u| δ. Without loss of generality, we can choose this δ > R 1 , where R 1 is defined in (H4). Therefore, we have On the other hand, when |w n | R 1 , from (H4) we have Consequently, from (7) we obtain Thus, given the arbitrariness of ε, there exists Λ > 0 such that w n → 0 in E when λ > Λ. This implies that u n → u in E, and Definition 1(i) holds. Finally, we prove that Definition 1(ii) holds. We argue indirectly, i.e., suppose that there exist c ∈ R and {u n } n∈N ⊂ E such that I(u n ) → c, u n → ∞, I (u n ) u n → 0 as n → ∞.
Then we have Using Lemma 1(iv), (13), and (g), we obtain Let v n = u n / u n . Then v n = 1, and there exists a function v ∈ E such that v n v weakly in E, v n → v strongly in L r (R 2 ) with r ∈ (2, +∞), v n (x) → v(x) for a.e.
x ∈ R 2 . Define a set Ω n (a, b) = {x ∈ R 2 : a |u n (x)| < b} with 0 a < b, and consider the following two possible cases.
Case 1. The function v is a zero function in E, i.e., v = 0, and v n 0 weakly in E, v n (x) → 0 for a.e. x ∈ R 2 . From (2) we have On the other hand, by the Hölder inequality, (14), and (H4) we obtain Combining (16) and (17), we have which contradicts (15). f (u n )u n |u n | 6 |v n | 6 dx This is also a contradiction.
Combining the above two cases, we have that Definition 1(ii) holds. Thus, I satisfies the Cerami condition (C). This completes the proof.
Proof of Theorem 1. Note that E is a Hilbert space, and let e j be an orthonomormal basis of E. Then we have In what follows, we show that for each k ∈ N, there exist ρ k > r k > 0 such that and a k = max u∈Y k , u =ρ k I(u) 0.
Thus, (19) holds. Finally, (H5) implies that I is an even functional on E, and by Lemma 4 I satisfies all the conditions of Lemma 2. Then I has a sequence of critical points {u n } such that I(u n ) → +∞ as n → ∞. This means that (1) has infinitely many high energy solutions. This completes the proof.