Infinitely Many High Energy Solutions for the Generalized Chern-Simons-Schrödinger System

(V1) VðxÞ ∈ CðR2Þ, VðxÞ =Vð∣x ∣ Þ, and VðxÞ ≥ 0 onR2; (V2) There exists b > 0 such that Vb = fx ∈R2 : VðxÞ < bg is nonempty and has finite measure; (V3) There exists R > 0 such that Bð0, RÞ = int V−1ð0Þ and Bð0, RÞ =V−1ð0Þ, where Bð0, RÞ denotes the ball of radius R centered at 0; (H1) f ∈ CðR,RÞ, and f ðuÞ = oð∣u ∣ Þ as ∣u ∣⟶0; (H2) There exists R0 ≥ 0 such that FðuÞ = Ð u 0 f ðtÞdt ≥ 0 and FðuÞ = ð1/6Þf ðuÞu − FðuÞ ≥ 0 for ∣u ∣ ≥R0; (H3) ð f ðuÞuÞ/juj6 ⟶ +∞ as ∣u ∣⟶∞;

In [16], the authors studied the following nonhomogeneous Schrödinger-Kirchhoff-type fourth-order Elliptic equations in ℝ N : They obtained the existence of infinitely many solutions for this system by means of the symmetry mountain pass theorem and the fountain theorem. Now, we state the main result: Then, for arbitrarily small μ > 0, there exists λ 0 > 0 such that system (1) possesses infinitely many high energy solutions when λ ≥ λ 0 . On the other hand, using (H1) for all ε > 0, we have |f ðuÞ | ≤ε juj, for|u| ≤ R 2 . Therefore, we obtain and thus,

Preliminaries
Let ∥·∥ s be the usual L s -norm for s ∈ ½1,+∞Þ, and l i ,l i ði ∈ ℝ + Þ stand for different positive constants. We use H 1 r ðℝ 2 Þ to denote a Sobolev space with the norm Define the space Journal of Function Spaces with the inner product and norm Note the large parameter λ in Theorem 1, so we need the following inner product and norm: Define E λ = ðE,∥ · ∥ λ Þ; then, we have E λ which is a Hilbert space. Using (V1)-(V3), there exist positive constants λ 0 , l 0 (independent of λ) such that Moreover, by [28], the embedding E λ°L p ðℝ 2 Þ is continuous for p ∈ ½2,+∞Þ, and E λ ↪L p ðℝ 2 Þ is compact for p ∈ ð2,+ ∞Þ, i.e., there exists l p > 0 such that For convenience, let l p l 0 =l p . Now, on E λ , we define the following energy functional: By (V1)-(V3), (9), (10), and [8], I is of class C 1 ðE λ , ℝÞ, and Note that (16) in [3], we have Consequently, we have Lemma 3 (see [4,7,8]). Suppose that u n converges to u a.e. in ℝ 2 and u n converges weakly to u in H 1 r ðℝ 2 Þ: Let A α,n ≔ A α ðu n ðxÞÞ, α = 0, 1, 2: Then, Ð ℝ 2 A 2 i,n u n udx, We say that I ∈ C 1 ðX, ℝÞ satisfies ðCÞ c -condition if any sequence fu n g such that has a convergent subsequence.
Lemma 4 (see [29]). Suppose that X is an infinite dimensional Banach space, and Y, Z are two subspaces of X with

Main Results
In order to prove Theorem 1, we provide some lemmas.
is bounded in E λ .
Proof. To prove the boundedness of fu n g, argument by contrary, assume that ∥u n ∥ λ ⟶ ∞. Let v n = u n /ku n k λ . Then, kv n k λ = 1, and kv n k p ≤l p kv n k λ =l p , p ≥ 2. Note that 3 Journal of Function Spaces qq ′ /ðq ′ − 1Þ > 2 by (g); for large n, from (16), we have for the fact that q ∈ ð1, 2Þ and μ > 0 is an arbitrarily small parameter.
In view of (20), we have Recall that kv n k λ = 1, and there exists a function v ∈ E λ such that v n ⇀ v weakly in E λ , v n → v strongly in L r ðℝ 2 Þ with r ∈ ð2,+∞Þ and v n ðxÞ → vðxÞ for a.e. x ∈ ℝ 2 . Define a set Ω n ða, bÞ = fx ∈ ℝ 2 : a≤|u n ðxÞ|<bg with 0 ≤ a < b, and we consider the following two possible cases. Case 1. v = 0, and v n ⇀ 0 weakly in E λ , v n ðxÞ ⟶ 0 for a.e.