1 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 R2:

(1)

We first list our assumptions for our problem (1):

(V1) V ∈ C(R2, R), and infx∈R2 V (x) ≥ V0 > 0, where V0 is a positive constant.

(V2) There exists b > 0 such that meas {x ϵ R2: V (x) ≤ b} is finite; here meas denotes the Lebesgue measure.

(H1) f ∈ C(R, R), and f (u) = o(|u|) as |u| → 0.

(H2) There exists R0 ≥ 0 such that F (u) = ∫u0 f (t) dt ≥ 0 and F(u) = f (u)u/6 – u F (u) ≥ 0 for |u| ≥ R0.

(H3) f (u)u/ |u|6 → +∞ as |u| → ∞.

(H4) There exist α0, R1 > 0, and τ ∈ (1, +∞) such that |f (u)|τ ≤ α0F(u)|u|τ for |u| ≥ R1.

(H5) f (−u) = −f (u) for u ∈ R.

(g) g ∈ Lq' (R2), and g(x) ≥ 0 (/≡ 0) for x ∈ R2, where q′ ∈ (1, 2/(2 − q)), q ∈ (1, 2).

Problem (1) arises in the study of standing wave solutions for the gauged nonlinear Schrödinger equation

where i denotes the imaginary unit, ∂0 = ∂/∂t, ∂1 = ∂/∂x1, ∂2 = ∂/∂x2 for (t, x1, x2) R1+2, φ : R1+2 → C is the complex scalar field, Aκ : R1+2 → R is the gauge field, and Dκ = ∂κ + iAκ is the covariant derivative for κ = 0, 1, 2. From the initial study in [8, 9] many papers on this system appeared in the literature; we refer the reader to [1, 2, 4,5,6,7,8, 10, 11, 13, 14, 18,19,20,21, 25, 26, 28, 29] and the references therein.

When λ = 1, the authors [12] obtained the existence and multiplicity of solutions for (1) with concave-convex nonlinearities µg(x, u) + νf (x, u), where g has sublinear growth, and f has asymptotically linear or superlinear growth. In [20], the authors studied the existence, nonexistence, and multiplicity of standing waves for (1) (λ = 1, µ = 0) with asymptotically linear nonlinearities and external potential, and in [1, 2, 4,5,6,7, 11, 13, 14, 18, 19, 21, 25, 26, 28, 29], the authors studied the existence and multiplicity of solutions (including sign-changing solutions and ground state solutions) for gauged nonlinear Schrödinger equation

Moreover, in [26], the authors also discussed the energy doubling property, i.e., the energy of sign-changing solutions is strictly larger than two times the least energy. In [10], the authors studied the existence and multiplicity of the positive standing wave with f (u) + ϵk(x), where the nonlinearity f behaves like exp(α │u│ 2) as │u│→∞. Moreover, they obtained a mountain-pass type solution when ϵ = 0.

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:

where ( -∆)s 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.

Remark 1. By virtue of (H1), (H2), and (H4) we can obtain a growth condition for f . Using (H2) and (H4), for |u| ≥ R2 := max{R0, R1}, we have

Let p = (τ + 1)/(τ − 1) + 1 = 2τ/(τ − 1). Then from (H4) we have p ∈ (2, +∞), and

On the other hand, using (H1), for all ε > 0, we have

.f (u). ≤ ε|u| for |u| ≤ R2.

Therefore, by the above two inequalities we have the growth condition for f :

(2)

Note the relation F and f , and we obtain

(3)

Remark 2. Let f (t) = t5(6 log |t| + 1), t ∈ R, and t 0. Then F (t) = t6 ln |t|, and we can check that f , F satisfy (H1)–(H5). For example, if we take τ ∈ (1, 3/2), we have

Consequently, for |t| large, we obtain

This implies that (H4) holds. Moreover, this function also satisfies (H1)–(H3) and (H5). However, this function does not satisfy the Ambrosetti–Rabinowitz condition, namely:

(AR) There exists µ > 6 such that 0 < µF (u) ≤ f (u)u for u ∈ R \ {0}.

2 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 ‹ → Lr(R2) is continuous for r ϵ [2, + ∞ ) and E ‹ → Lr(R2) is compact for r ϵ (2, + ∞ ), i.e., there are constants γr > 0 such that ǁuǁ r ≤ γr ǁuǁ for 2 ≤ r < ∞, where ǁ.ǁ r is the norm in the usual Lebesgue space Lr(R2).

In what follows, we present the energy functional I: E → R for problem (1) defined as

Where

Note (3) and (g). We obtain that I is of class C1 and its derivative is

where

Lemma 1. (See [1, 13, 14, 29].) Suppose that {un} converges weakly to a function u in E as n → ∞. Then

(i) limn→+∞ B(un) = B(u),

(ii) limn→+∞⟨B′(un), un⟩ = ⟨B′(u), u⟩,

(iii) limn→+∞⟨B′(un), ϕ⟩ = ⟨B′(u), ϕ⟩,

(iv) ⟨B′(u), u⟩ = 6B(u),

(v) B(u) ≤ C0ǁuǁ4ǁuǁ2 ≤ C0γ2γ4ǁuǁ6 := C1ǁuǁ6 for some C0, C1 > 0.

In order to obtain our main result, we need to introduce the Fountain theorem under the Cerami condition (C).

Definition 1. (See [16].) Assume that X is a Banach space. We say that J satisfies the Cerami condition if

(C) J ∈ C1(X, R), and for all c ∈ R,

(i) any bounded sequence {un} C X satisfying J(un) → c, J′(un) → 0 possesses a convergent subsequence;

(ii) there exist σ, R, β > 0 such that for any u ∈ J−1([c − σ, c + σ]) with ǁuǁ ≥ R, ǁJ′(u)ǁǁuǁ ≥ β.

Lemma 2. (See [16].) Assume that , where Xj are finite dimensional

subspaces of X. For each k ϵ N, let . Suppose that J ∈ C1(X, R) satisfies the Cerami condition (C) and J(−u) = J(u). Assume for each k ∈ N, there exist ρk > rk > 0 such that

(i) bk = infu∈Zk ∩Srk J(u) → +∞, k → ∞,

(ii) ak = maxu∈Yk ∩Sρk J(u) ≤ 0, where Sρ = {u ∈ X: ǁuǁ = ρ}.

Then J has a sequence of critical points un such that J(un) → +∞ as n → ∞.

3 Proof of Theorem 1

Lemma 3. Let sequence {un} converge weakly to a function u in E, un(x) → u(x) a.e. in R2 as n → ∞. Then

(4)

(5)

In particular, if

then

(6)

and

(7)

Proof. From the compactness of E ‹→ Lr(R2), for r ∈ (2, +∞), we have

Let wn = un − u. Then we have

Since un u in E, we have (un − u, u) → 0 as n → ∞, which implies

ǁunǁ2 = (wn + u, wn + u) = ǁwnǁ2 + ǁuǁ2 + o(1) as n → ∞.

Note Lemma 1(v), and we have B(un − u) ≤ C0ǁun − uǁ4ǁun − uǁ2 → 0 as n → ∞.

Consequently, to obtain (4), by Lemma 1(i) we only need to check that

(8)

And

(9)

Note the definition of ( , ), for all n ϵ N, we have (un, ϕ) = (un - u, ϕ) + (u, ϕ). Moreover, since wn 0 in E and by Lemma 1(iii), to prove (5), it suffices to show that

(10)

And

(11)

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 Cq > 0 such that ǁun│ q−2un - │u q−2u│ ≤ Cq │un - u│ q−1. Therefore, from (g) and the Hölder inequality we only need to prove:

Consequently, (11) is true. Note that we can use similar methods in Lemma 4.7 of [30] to prove (10). In what follows, we prove (8). Using the ideas in [17, 22, 23], we have

Hence, from (2) we obtain

for some ε1, Cε1 > 0, where p > 2. Therefore, together with (3), using the Young inequality with ε (for all ε > 0), we obtain

Consequently, we consider the function fn defined as

Then

and by the Lebesgue dominated convergence theorem we have

(12)

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 (un − u, ϕ) → 0 as n → ∞. It suffices to check that ∫R2 f(wn) ᵩ dx = 0(1) as n → ∞. Note the arbitrariness of ϵ in (2), and wn → 0 in Lp(R2), p > 2. Therefore, from (2) we have

This completes the proof.

Lemma 4. Suppose that all the assumptions in Theorem 1 hold. Then I satisfies the Cerami condition (C).

Proof. For all c ∈ R, suppose that there exists {un}n∈N ⊂ E is bounded and

I(un) → c, I′(un) → 0 as n → ∞.

Using ⟨I′(u), ϕ⟩ = 0 for all ϕ ∈ E in Lemma 3 and noting Lemma 1(iv), we have

This implies that

Recall wn = un − u. From (6) and (7) we have

As V (x) < b on a set of finite measure and wn → 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 ϵ R2 and u ≤ δ. Without loss of generality, we can choose this δ > R1, where R1 is defined in (H4). Therefore, we have

On the other hand, when |wn| ≥ R1, from (H4) we have

Consequently, from (7) we obtain

Thus, given the arbitrariness of ε, there exists Λ > 0 such that wn 0 in E when λ > Λ.

This implies that un → 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 {un}n∈N ⊂ E such that

(13)

Then we have

(14)

Using Lemma 1(iv), (13), and (g), we obtain

(15)

Let vn = un/ ǁunǁ . Then ǁvnǁ = 1, and there exists a function v E such that vn → v weakly in E, vn → v strongly in Lr(R2) with r ϵ (2, + ∞ ), vn(x) → v(x) for a.e. x ϵ R2. Define a set Ωn(a, b) = {x ϵ R2: a ≤ un(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 vn 0 weakly in E, vn(x) → 0 for a.e. x ∈ R2. From (2) we have

(16)

On the other hand, by the Hölder inequality, (14), and (H4) we obtain

(17)

Combining (16) and (17), we have

which contradicts (15).

Case 2. The function v is not a zero function in E, i.e., v(x) ≡ 0, x ∈ R2. Hence, if we set A = {x ∈ R2: v(x) =/ 0}, then meas A > 0. For x ∈ A, we have limn→∞ un(x) = ∞, and hence A C Ωn(R1, ∞) for large n. By (H3) and Lemma 1(iv), (v), noting the nonnegativity of f (u)u, Fatou’s Lemma enables us to obtain

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 ej be an orthonomormal basis of E. Then we have

In what follows, we show that for each k ∈ N, there exist ρk > rk > 0 such that

(18)

and

(19)

Note that the compact embedding E ‹→ Lr(R2) with r ∈ (2, +∞), and by Lemma 3.8 in [24] we have βk(r) = supu Zk, ǁuǁ =1 ǁuǁ r → 0, k→∞ . This, together with (3), implies that

(20)

Note that p = 2τ/(τ − 1), and if we take ε ≤ 1/(γ22(2τ − 1)) and rk = (cεβkp)1/(2−p), by (20), for u ∈ Zk and ǁuǁ = rk, we find

with τ > 1, p > 2. Therefore, (18) holds.

On the other hand, for any finite dimensional subspace E ⊂ E, we show that

(21)

Arguing indirectly, assume that for some sequence {un} ⊂ E with ǁunǁ → ∞, there exists M > 0 such that I(un) ≥ −M for all n ∈ N. Let vn = un/ǁunǁ. Then ǁvnǁ = 1, and there is a function v ∈ E such that vn v in E. Since dim E < ∞, we have vn → v in E˜, vn(x) → v(x) for a.e. x ∈ R2, and ǁvǁ = 1. Let Ω = {x ∈ R2: v(x) /= 0}. Then meas Ω > 0, and limn→∞ |un(x)| → ∞ for a.e. x ∈ Ω. From Lemma 1(v) and (g) we have

(22)

From the L’Hôspital’s rule and (H3) we have

Fatou’s lemma implies that

This contradicts (22), and thus (21) holds. As a result, we can take u Yk and large ρk (ρk > rk) such that

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 un such that (un) →+ ∞ as n →∞. This means that (1) has infinitely many high energy solutions.

This completes the proof.

Acknowledgments

The authors would like to thank the referee for his/her valuable comments and suggestions.

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Notes