Stochastic homogenization of maximal monotone relations and applications

1.   Introduction

In this paper, we discuss the existence and multiplicity of standing wave solutions for the following perturbed fractional p-Laplacian systems with critical nonlinearity

where $ \varepsilon $ is a positive parameter, $ N > ps, s\in (0, 1), p^{*}_{s} = \frac{Np}{N-ps} $ and $ (-\Delta)^{s}_{p} $ is the fractional p-Laplacian operator, which is defined as

where $ B_{\varepsilon}(x) = \{y\in \mathbb{R}^{N}: |x-y| < \varepsilon\} $. The functions $ V(x), K(x) $ and $ F(x, u, v) $ satisfy the following conditions:

$(V_{0})\; V\in C(\mathbb{R}^{N}, \mathbb{R}), \min_{x\in \mathbb{R}^{N}} V(x) = 0 {\rm{\; and\; there\; is\; a\; constant}}\; b > 0 {\rm{\; such \; that\; the\; set}}\; $ $ V^{b}: = \{x\in \mathbb{R}^{N}: V(x) < b\} {\rm{\; has\; finite\; Lebesgue\; measure}}; $

$(K_{0})\; K\in C(\mathbb{R}^{N}, \mathbb{R}), 0 < \inf K\leq \sup K < \infty; $

$(F_{1})\; F\in C^{1}(\mathbb{R}^{N} \times \mathbb{R}^{2}, \mathbb{R})\; {{\rm{and}}}\; F_{s}(x, s, t), F_{t}(x, s, t) = o(|s|^{p-1} + |t|^{p-1}) $ $ {\rm{uniformly\; in}}\; x\in \mathbb{R}^{N}\; {{\rm{as}}} \; |s|+|t|\rightarrow 0;$

$(F_{2})\; {\rm{there \; exist}}\; C_{0} > 0\; {\rm{ and}}\; p < \kappa < p_{s}^{*}\; {\rm{ such that}} $ $|F_{s}(x, s, t)|, |F_{t}(x, s, t)|\leq C_{0}(1+ |s|^{\kappa-1} + |t|^{\kappa-1}); $

$(F_{3})\; {\rm{there \; exist}}\; l_{0} > 0, \; d > p \; {\rm{and}}\; \mu\in (p, p_{s}^{*}) \; {\rm{such\; that}}\; F(x, s, t)\geq l_{0}(|s|^{d} +|t|^{d})\; {\rm{and}} $ $0 < \mu F(x, s, t)\leq F_{s}(x, s, t)s + F_{t}(x, s, t)t \; {{\rm{for \; all}}}\; (x, s, t)\in \mathbb{R}^{N}\times \mathbb{R}^{2};$

$(F_{4})\; F_{s}(x, -s, t) = -F_{s}(x, s, t)\; {{\rm{and}}}\; F_{t}(x, s, -t) = -F_{t}(x, s, t)\; {\rm{for\; all}}\; (x, s, t)\in \mathbb{R}^{N} \times \mathbb{R}^{2}. $

Conditions $ (V_{0}), (K_{0}) $, suggested by Ding and Lin ^[11] in studying perturbed Schrödinger equations with critical nonlinearity, and then was used in ^[28,32,33].

In recent years, a great deal of attention has been focused on the study of standing wave solutions for perturbed fractional Schrödinger equation

where $ s\in (0, 1) $, $ N > 2 $s and $ \varepsilon > 0 $ is a small parameter. It is well known that the solution of (1.2) is closely related to the existence of solitary wave solutions for the following eqation

where $ i $ is the imaginary unit. $ (-\Delta)^{s} $ is the fractional Laplacian operator which arises in many areas such as physics, phase transitions, chemical reaction in liquids, finance and so on, see ^{[1,6,18,22,27]}. Additionally, Eq (1.2) is a fundamental equation of fractional quantum mechanics. For more details, please see ^[17,18].

Equation (1.2) was also investigated extensively under various hypotheses on the potential and the nonlinearity. For example, Floer and Weinstein ^[12] first considered the existence of single-peak solutions for $ N = 1 $ and $ f(t) = t^{3} $. They obtained a single-peak solution which concentrates around any given nondegenerate critical point of $ V $. Jin, Liu and Zhang ^[16] constructed a localized bound-state solution concentrating around an isolated component of the positive minimum point of $ V $, when the nonlinear term $ f(u) $ is a general critical nonlinearity. More related results can be seen in ^{[5,7,10,13,14,26,43]} and references therein. Recently, Zhang and Zhang ^[46] obtained the multiplicity and concentration of positive solutions for a class of fractional unbalanced double-phase problems by topological and variational methods. Related to (1.2) with $ s = 1 $, see ^[31,39] for quasilinear Schrödinger equations.

On the other hand, fractional p-Laplacian operator can be regarded as an extension of fractional Laplacian operator. Many researchers consider the following equation

When $ f(x, u) = A(x)|u|^{p^{*}_{s}-2}u + h(x, u) $, Li and Yang ^[21] obtained the existence and multiplicity of weak solutions by variational methods. When $ f(x, u) = \lambda f(x)|u|^{q-2}u + g(x)|u|^{r-2}u $, under suitable assumptions on nonlinearity and weight functions, Lou and Luo ^[19] established the existence and multiplicity of positive solutions via variational methods. With regard to the p-fractional Schrödinger-Kirchhoff, Song and Shi ^[29] considered the following equation with electromagnetic fields

They obtained the existence and multiplicity solutions for (1.4) by using the fractional version of concentration compactness principle and variational methods, see also ^{[24,25,34,35,38,41]} and references therein. Related to (1.3) with $ s = 1 $, see ^[15,23].

Recently, from a mathematical point of view, (fractional) elliptic systems have been the focus for many researchers, see ^{[2,8,9,20,30,37,42,44,45]}. As far as we know, there are few results concerned with the (fractional) p-Laplacian systems with a small parameter. In this direction, we cite the work of Zhang and Liu ^[40], who studied the following p-Laplacian elliptic systems

By using variational methods, they proved the existence of nontrivial solutions for (1.5) provided that $ \varepsilon $ is small enough. In ^[36], Xiang, Zhang and Wei investigated the following fractional p-Laplacian systems without a small parameter

Under some suitable conditions, they obtained the existence of nontrivial and nonnegative solutions for (1.6) by using the mountain pass theorem.

Motivated by the aforementioned works, it is natural to ask whether system (1.5) has a nontrivial solution when the p-Laplacian operator is replaced by the fractional p-Laplacian operator. As far as we know, there is no related work in this direction so far. In this paper, we give an affirmative answer to this question considering the existence and multiplicity of standing wave solutions for (1.1).

Now, we present our results of this paper.

Theorem 1.1. Assume that $ (V_{0}) $, $ (K_{0}) $ and $ (F_{1}) $–$ (F_{3}) $ hold. Then for any $ \tau > 0 $, there is $ \Gamma_{\tau} > 0 $ such that if $ \varepsilon < \Gamma_{\tau} $, system (1.1) has at least one solution $ (u_{\varepsilon}, v_{\varepsilon})\rightarrow (0, 0) $ in $ W $ as $ \varepsilon\rightarrow 0 $, where $ W $ is stated later, satisfying:

and

Theorem 1.2. Let $ (V_{0}) $, $ (K_{0}) $ and $ (F_{1}) $–$ (F_{4}) $ hold. Then for any $ m\in \mathbb{N} $ and $ \tau > 0 $ there is $ \Gamma_{m\tau} > 0 $ such that if $ \varepsilon < \Gamma_{m\tau} $, system (1.1) has at least $ m $ pairs of solutions $ (u_{\varepsilon}, v_{\varepsilon}) $, which also satisfy the above estimates in Theorem 1.1. Moreover, $ (u_{\varepsilon}, v_{\varepsilon}) \rightarrow (0, 0) $ in $ W $ as $ \varepsilon\rightarrow 0 $.

Remark 1.1. On one hand, our results extend the results in ^[40], in which the authors considered the existence of solutions for perturbed $ p $-Laplacian system, i.e., system (1.1) with $ s = 1 $. On the other hand, our results also extend the results in ^[21] to a class of perturbed fractional $ p $-Laplacian system (1.1).

Remark 1.2. Compared with the results obtained by ^{[12,13,14,15,16]}, when $ \varepsilon\rightarrow0 $, the solutions of Theorems 1.1 and 1.2 are close to trivial solutions.

In this paper, our goal is to prove the existence and multiplicity of standing wave solutions for (1.1) by variational approach. The main difficulty lies on the lack of compactness of the energy functional associated to system (1.1) because of unbounded domain $ \mathbb{R}^{N} $ and critical nonlinearity. To overcome this difficulty, we adopt some ideas used in ^[11] to prove that $ (PS)_{c} $ condition holds.

The rest of this article is organized as follows. In Section 2, we introduce the working space and restate the system in a equivalent form by replacing $ \varepsilon^{-ps} $ with $ \lambda $. In Section 3, we study the behavior of $ (PS)_{c} $ sequence. In Section 4, we complete the proof of Theorems 2.1 and 2.2, respectively.

2.   Preliminaries

To obtain the existence and multiplicity of standing wave solutions of system (1.1) for small $ \varepsilon $, we rewrite (1.1) in a equivalent form. Let $ \lambda = \varepsilon^{-ps} $, then system (1.1) can be expressed as

for $ \lambda\rightarrow +\infty $.

We introduce the usual fractional Sobolev space

equipped with the norm

where $ |\cdot|_{p} $ is the norm in $ L^{p}(\mathbb{R}^{N}) $ and

is the Gagliardo seminorm of a measurable function $ u: \mathbb{R}^{N}\rightarrow \mathbb{R} $. In this paper, we continue to work in the following subspace of $ W^{s, p}(\mathbb{R}^{N}) $ which is defined by

with the norm

Notice that the norm $ ||\cdot||_{s, p} $ is equivalent to $ ||\cdot||_{\lambda} $ for each $ \lambda > 0 $. It follows from $ (V_{0}) $ that $ W_{\lambda} $ continuously embeds in $ W^{s, p}(\mathbb{R}^{N}) $. For the fractional system (2.1), we shall work in the product space $ W = W_{\lambda}\times W_{\lambda} $ with the norm $ ||(u, v)||^{p} = ||u||^{p}_{\lambda} +||v||^{p}_{\lambda} $ for any $ (u, v)\in W $.

We recall that $ (u, v)\in W $ is a weak solution of system (2.1) if

for all $ (\phi, \psi)\in W $.

Note that the energy functional associated with (2.1) is defined by

Clearly, it is easy to check that $ \Phi_{\lambda}\in C^{1}(W, \mathbb{R}) $ and its critical points are weak solution of system (2.1).

In order to prove Theorem 1.1 and 1.2, we only need to prove the following results.

Theorem 2.1. Assume that $ (V_{0}) $, $ (K_{0}) $ and $ (F_{1}) $–$ (F_{3}) $ hold. Then for any $ \tau > 0 $, there is $ \Lambda_{\tau} > 0 $ such that if $ \lambda\geq\Lambda_{\tau} $, system (2.1) has at least one solution $ (u_{\lambda}, v_{\lambda})\rightarrow (0, 0) $ in $ W $ as $ \lambda\rightarrow \infty $, satisfying:

and

Theorem 2.2. Assume that $ (V_{0}) $, $ (K_{0}) $ and $ (F_{1}) $–$ (F_{4}) $ hold. Then for any $ m\in \mathbb{N} $ and $ \tau > 0 $ there is $ \Lambda_{m\tau} > 0 $ such that if $ \lambda\geq\Lambda_{m\tau} $, system (2.1) has at least $ m $ pairs of solutions $ (u_{\lambda}, v_{\lambda}) $, which also satisfy the estimates in Theorem 2.1. Moreover, $ (u_{\lambda}, v_{\lambda}) \rightarrow (0, 0) $ in $ W $ as $ \lambda\rightarrow \infty $.

3.   Behaviors of $ (PS)_{c} $ sequences

In this section, we are focused on the compactness of the functional $ \Phi_{\lambda} $.

Recall that a sequence $ \{(u_{n}, v_{n})\}\subset W $ is a $ (PS)_{c} $ sequence at level $ c $, if $ \Phi_{\lambda}(u_{n}, v_{n})\rightarrow c $ and $ \Phi'_{\lambda}(u_{n}, v_{n})\rightarrow 0 $. $ \Phi_{\lambda} $ is said to satisfy the $ (PS)_{c} $ condition if any $ (PS)_{c} $ sequence contains a convergent subsequence.

Proposition 3.1. Assume that the conditions $ (V_{0}), (K_{0}) $ and $ (F_{1}) $–$ (F_{3}) $ hold. Then there exists a constant $ \alpha > 0 $ independent of $ \lambda $ such that, for any $ (PS)_{c} $ sequence $ \{(u_{n}, v_{n})\}\subset W $ for $ \Phi_{\lambda} $ with $ (u_{n}, v_{n})\rightharpoonup (u, v) $, either $ (u_{n}, v_{n})\rightarrow (u, v) $ or $ c - \Phi_{\lambda}(u, v) \geq \alpha\lambda^{1-\frac{N}{ps}} $.

Corollary 3.1. Under the assumptions of Proposition 3.1, $ \Phi_{\lambda} $ satisfies the $ (PS)_{c} $ condition for all $ c < \alpha\lambda^{1-\frac{N}{ps}} $.

The proof of Proposition 3.1 consists of a series of lemmas which will occupy the rest of this section.

Lemma 3.1. Assume that $ (V_{0}), (K_{0}) $ and $ (F_{3}) $ are satisfied. Let $ \{(u_{n}, v_{n})\}\subset W $ be a $ (PS)_{c} $ sequence for $ \Phi_{\lambda} $. Then $ c\geq 0 $ and $ \{(u_{n}, v_{n})\} $ is bounded in $ W $.

Proof. Let $ \{(u_{n}, v_{n})\} $ be a $ (PS)_{c} $ sequence for $ \Phi_{\lambda} $, we obtain that

By $ (K_{0}) $ and $ (F_{3}) $, we deduce that

which implies that there exists $ M > 0 $ such that

Thus, $ \{(u_{n}, v_{n})\} $ is bounded in $ W $. Taking the limit in (3.1), we show that $ c\geq0 $. This completes the proof.

From the above lemma, there exists $ (u, v)\in W $ such that $ (u_{n}, v_{n})\rightharpoonup (u, v) $ in $ W $. Furthermore, passing to a subsequence, we have $ u_{n}\rightarrow u $ and $ v_{n}\rightarrow v $ in $ L^{\gamma}_{loc}(\mathbb{R}^{N}) $ for any $ \gamma\in [p, p_{s}^{*}) $ and $ u_{n}(x)\rightarrow u(x) $ and $ v_{n}(x)\rightarrow v(x) $ a.e. in $ \mathbb{R}^{N} $. Clearly, $ (u, v) $ is a critical point of $ \Phi_{\lambda} $.

Lemma 3.2. Let $ \{(u_{n}, v_{n})\} $ be stated as in Lemma 3.1 and $ \gamma\in [p, p_{s}^{*}) $. Then there exists a subsequence $ \{(u_{n_{j}}, v_{n_{j}})\} $ such that for any $ \varepsilon > 0 $, there is $ r_{\varepsilon} > 0 $ with

for all $ r\geq r_{\varepsilon} $, where, $ B_{r}: = \{x\in \mathbb{R}^{N}: |x|\leq r\} $.

Proof. The proof is similar to the one of Lemma 3.2 of ^[11]. We omit it here.

Let $ \sigma: [0, \infty)\rightarrow [0, 1] $ be a smooth function satisfying $ \sigma(t) = 1 $ if $ t\leq 1 $, $ \sigma(t) = 0 $ if $ t\geq2 $. Define $ \overline{u}_{j}(x) = \sigma(\frac{2|x|}{j})u(x) $, $ \overline{v}_{j}(x) = \sigma(\frac{2|x|}{j})v(x) $. It is clear that

Lemma 3.3. Let $ \{(u_{n_{j}}, v_{n_{j}})\} $ be stated as in Lemma 3.2, then

and

uniformly in $ (\phi, \psi)\in W $ with $ ||(\phi, \psi)||\leq 1 $.

Proof. By (3.2) and the local compactness of Sobolev embedding, we know that for any $ r > 0 $,

uniformly for $ ||\phi||\leq 1 $. For any $ \varepsilon > 0 $, there exists $ r_{\varepsilon} > 0 $ such that

for all $ r\geq r_{\varepsilon} $, see [Lemma 3.2, 11]. From $ (F_{1}) $ and $ (F_{2}) $, we obtain

Thus, from (3.3), (3.4) and the Hölder inequality, we have

where $ C_{1}, C_{2}, C_{3} $ and $ C_{4} $ are positive constants. Similarly, we can deduce that the other equality also holds.

Lemma 3.4. Let $ \{(u_{n_{j}}, v_{n_{j}})\} $ be stated as in Lemma 3.2, the following facts hold:

$ (i)\; \Phi_{\lambda} (u_{n_{j}} - \overline{u}_{j}, v_{n_{j}} - \overline{v}_{{j}})\rightarrow c - \Phi_{\lambda}(u, v); $

$(ii)\; \Phi'_{\lambda} (u_{n_{j}} - \overline{u}_{j}, v_{n_{j}} - \overline{v}_{{j}}) \rightarrow 0 \; {\rm{in}} \; W^{-1}\; ({\rm{the \; dual\; space\; of\; W}}). $

Proof. $ (i) $ We have

Using (3.2) and the Brézis-Lieb Lemma ^[4], it is easy to get

and

Using the fact that $ \Phi_{\lambda} (u_{n_{j}}, v_{n_{j}}) \rightarrow c $ and $ \Phi_{\lambda} (\overline{u}_{j}, \overline{v}_{{j}})\rightarrow \Phi_{\lambda}(u, v) $ as $ j\rightarrow \infty $, we have

$ (ii) $ We observe that for any $ (\phi, \psi)\in W $ satisfying $ ||(\phi, \psi)||\leq 1 $,

It follows from a standard argument that

and

uniformly in $ ||(\phi, \psi)||\leq1 $. By Lemma 3.3, we obtain $ \Phi'_{\lambda} (u_{n_{j}} - \overline{u}_{j}, v_{n_{j}} - \overline{v}_{{j}}) \rightarrow 0 $. We complete this proof.

Set $ u^{1}_{j} = u_{n_{j}} - \overline{u}_{j} $, $ v^{1}_{j} = v_{n_{j}} - \overline{v}_{j} $, then $ u_{n_{j}} -u = u^{1}_{j} + (\overline{u}_{j} - u) $, $ v_{n_{j}} -v = v^{1}_{j} + (\overline{v}_{j} - v) $. From (3.2), we have $ (u_{n_{j}}, v_{n_{j}})\rightarrow (u, v) $ if and only if $ (u^{1}_{j}, v^{1}_{j})\rightarrow (0, 0) $. By Lemma 3.4, one has along a subsequence that $ \Phi_{\lambda}(u^{1}_{j}, v^{1}_{j}) \rightarrow c -\Phi_{\lambda}(u, v) $ and $ \Phi'_{\lambda}(u^{1}_{j}, v^{1}_{j})\rightarrow 0 $.

Note that $ \langle \Phi'_{\lambda}(u^{1}_{j}, v^{1}_{j}), (u^{1}_{j}, v^{1}_{j}) \rangle = 0 $, by computation, we get

Hence, by $ (F_{3}) $ and (3.5), we have

where $ K_{min} = \inf_{x\in \mathbb{R}^{N}}K(x) > 0 $. So, it is easy to see that

Denote $ V_{b}(x) = \max \{V(x), b\} $, where $ b $ is the positive constant from assumption of $ (V_{0}) $. Since the set $ V^{b} $ has finite measure and $ (u^{1}_{j}, v^{1}_{j})\rightarrow (0, 0) $ in $ L^{p}_{loc}\times L^{p}_{loc} $, we obtain

By $ (K_{0}), (F_{1}) $ and $ (F_{2}) $, we can find a constant $ C_{b} > 0 $ such that

Let $ S $ is fractional Sobolev constant which is defined by

Proof of Proposition 3.1. Assume that $ (u_{n_{j}}, v_{n_{j}})\nrightarrow(u, v) $, then $ \lim\inf_{j\rightarrow \infty}||(u^{1}_{j}, v^{1}_{j})|| > 0 $ and $ c -\Phi_{\lambda}(u, v) > 0 $.

From (3.5), (3.7), (3.8) and (3.9), we deduce

Thus, by (3.6), we have

or equivalently

where $ \alpha = \frac{s K_{min}}{N}(\frac{S}{C_{b}})^{\frac{N}{ps}} $. The proof is complete.

4.   Proof of the main results

Lemma 4.1. Suppose that $ (V_{0}) $, $ (K_{0}), (F_{1}), (F_{2}) $ and $ (F_{3}) $ are satisfied, then the functional $ \Phi_{\lambda} $ satisfies the following mountain pass geometry structure:

$ (i) $ there exist positive constants $ \rho $ and $ a $ such that $ \Phi_{\lambda}(u, v)\geq a $ for $ ||(u, v)|| = \rho $;

$ (ii) $ for any finite-dimensional subspace $ Y\subset W $,

$ (iii) $ for any $ \tau > 0 $ there exists $ \Lambda_{\tau} > 0 $ such that each $ \lambda\geq \Lambda_{\tau} $, there exists $ \widetilde{\omega}_{\lambda}\in Y $ with $ ||\widetilde{\omega}_{\lambda}|| > \rho $, $ \Phi_{\lambda}(\widetilde{\omega}_{\lambda})\leq 0 $ and

Proof. $ (i) $ From $ (F_{1}), (F_{2}) $, we have for any $ \varepsilon > 0 $, there is $ C_{\varepsilon} > 0 $ such that

Thus, combining with (4.1) and Sobolev inequality, we deduce that

where $ \varepsilon $ is small enough and $ C_{5}, C_{6} > 0 $, thus $ (i) $ is proved because $ p_{s}^{*} > p $.

$ (ii) $ By $ (F_{3}) $, we define the functional $ \Psi_{\lambda}\in C^{1}(W, \mathbb{R}) $ by

Then

For any finite-dimensional subspace $ Y\subset W $, we only need to prove

In fact, we have

Since all norms in a finite dimensional space are equivalent and $ p < d < p_{s}^{*} $, thus $ (ii) $ holds.

$ (iii) $ From Corollary 3.1, for $ \lambda $ large and $ c $ small enough, $ \Phi_{\lambda} $ satisfies $ (PS)_{c} $ condition. Thus, we will find a special finite dimensional-subspace by which we construct sufficiently small minimax levels for $ \Phi_{\lambda} $ when $ \lambda $ large enough.

Recall that

see ^[40] for this proof. For any $ 0 < \varepsilon < 1 $, we can take $ \varphi_{\varepsilon}\in C^{\infty}_{0}(\mathbb{R}^{N}) $ with $ |\varphi_{\varepsilon}|_{d} = 1 $, supp $ \varphi_{\varepsilon}\subset B_{r_{\varepsilon}}(0) $ and $ [\varphi_{\varepsilon}]^{p}_{p, s} < \varepsilon $.

Let

For $ t\geq 0 $, $ (F_{3}) $ imply that

Indeed, for $ t > 0 $, define

It is easy to show that $ t_{0} = (\frac{\int\int_{\mathbb{R}^{2N}}\frac{|\varphi_{\varepsilon}(x)-\;\varphi_{\varepsilon}(y)|\;^{p}\;}{|x-y|^{N+ps}}\;\;\;dxdy + \int_{\mathbb{R}^{N}} V(\lambda^{-\frac{1}{ps}}x)\;|\;\varphi_{\varepsilon}\;|^{p} dx}{l_{0}d})^{\frac{1}{d-p}} $ is a maximum point of $ g $ and

Since $ V(0) = 0 $ and supp $ \varphi_{\varepsilon} \subset B_{r_{\varepsilon}}(0) $, there exists $ \Lambda_{\varepsilon} > 0 $ such that

Hence, we have

Choose $ \varepsilon > 0 $ such that

and taking $ \Lambda_{\tau} = \Lambda_{\varepsilon} $, from $ (ii) $, we can take $ \overline{t} $ large enough and define $ \widetilde{\omega}_{\lambda} = \overline{t}\overline{\omega}_{\lambda} $, then we have

Proof of Theorem 2.1. From Lemma 4.1, for any $ 0 < \tau < \alpha $, there exists $ \Lambda_{\tau} > 0 $ such that for $ \lambda\geq \Lambda_{\tau} $, we have

where $ \Gamma_{\lambda} = \{\eta\in C([0, 1], W): \eta(0) = 0, \eta(1) = \widetilde{\omega}_{\lambda}\} $. Furthermore, in virtue of Corollary 3.1, we obtain that $ (PS)_{c} $ condition hold for $ \Phi_{\lambda} $ at $ c $. Therefore, by the mountain pass theorem, there is $ (u_{\lambda}, v_{\lambda})\in W $ such that $ \Phi'_{\lambda}(u_{\lambda}, v_{\lambda}) = 0 $ and $ \Phi_{\lambda}(u_{\lambda}, v_{\lambda}) = c $.

Finally, we prove that $ (u_{\lambda}, v_{\lambda}) $ satisfies the estimates in Theorem 2.1.

Since $ (u_{\lambda}, v_{\lambda}) $ is a critical point of $ \Phi_{\lambda} $, there holds for $ \theta\in [p, p_{s}^{*}] $

Taking $ \theta = \mu $, we get the estimate (2.2) and taking $ \theta = p $ yields the estimate (2.3).

To obtain the multiplicity of critical points, we will adopt the index theory defined by the Krasnoselski genus.

Proof of Theorem 2.2. Denote the set of all symmetric (in the sense that $ -A = A $) and closed subsets of $ A $ by $ \sum $. For any $ A\in\sum $ let gen $ (A) $ be the Krasnoselski genus and

where $ \Upsilon $ is the set of all odd homeomorphisms $ k\in C(W, W) $ and $ \rho $ is the number from Lemma 4.1. Then $ i $ is a version of Benci's pseudoindex ^[3]. $ (F_{4}) $ implies that $ \Phi_{\lambda} $ is even. Set

If $ c_{\lambda_{j}} $ is finite and $ \Phi_{\lambda} $ satisfies $ (PS)_{c_{\lambda_{j}}} $ condition, then we know that all $ c_{\lambda_{j}} $ are critical values for $ \Phi_{\lambda} $.

Step 1. We show that $ \Phi_{\lambda} $ satisfies $ (PS)_{c_{\lambda_{j}}} $ condition at all levels $ c_{\lambda_{j}} < \tau\lambda^{1- \frac{N}{ps}} $.

To complete the claim, we need to estimate the level $ c_{\lambda_{j}} $ in special finite-dimensional subspaces.

Similar to proof in Lemma 4.1, for any $ m\in \mathbb{N} $, $ \varepsilon > 0 $ and $ j = 1, 2, \cdot \cdot \cdot, m $, one can choose $ m $ functions $ \varphi^{j}_{\varepsilon}\in C^{\infty}_{0}(\mathbb{R}^{N}) $ with supp $ \varphi^{i}_{\varepsilon}\bigcap $ supp $ \varphi^{j}_{\varepsilon} = \emptyset $ if $ i\neq j $, $ |\varphi^{j}_{\varepsilon}|_{d} = 1 $ and $ [\varphi^{j}_{\varepsilon}]^{p}_{p, s} < \varepsilon $.

Let $ r^{m}_{\varepsilon } > 0 $ be such that supp $ \varphi^{j}_{\varepsilon}\subset B_{r^{m}_{\varepsilon }}(0) $. Set

and define

Then $ i(F^{m}_{\lambda}) = \dim F^{m}_{\lambda} = m $. Observe that for each $ \widetilde{\omega} = \sum^{m}_{j = 1}t_{j}\overline{\omega}^{j}_{\lambda}\in F^{m}_{\lambda} $,

and for $ t_{j} > 0 $

Set

Since $ V(0) = 0 $ and supp $ \varphi^{j}_{\varepsilon} \subset B_{r^{m}_{\varepsilon }}(0) $, there exists $ \Lambda_{m \varepsilon } > 0 $ such that

Consequently, there holds

Choose $ \varepsilon > 0 $ small that $ ml_{0}(2\varepsilon)^{\frac{d}{d-p}} < \tau $. Thus for any $ m\in N $ and $ \tau\in(0, \alpha) $, there exists $ \Lambda_{m \tau } = \Lambda_{ m \varepsilon} $ such that $ \lambda > \Lambda_{m\tau } $, we can choose a $ m $-dimensional subspace $ F^{m}_{\lambda} $ with $ \max \Phi_{\lambda}(F^{m}_{\lambda}) \leq \tau \lambda^{1 - \frac{N}{ps}} $ and

From Corollary 3.1, we know that $ \Phi_{\lambda} $ satisfies the $ (PS) $ condition at all levels $ c_{\lambda_{j}} $. Then all $ c_{\lambda_{j}} $ are critical values.

Step 2. We prove that (2.1) has at least $ m $ pairs of solutions by the mountain-pass theorem.

By Lemma 4.1, we know that $ \Phi_{\lambda} $ satisfies the mountain pass geometry structure. From step 1, we note that $ \Phi_{\lambda} $ also satisfies $ (PS)_{c_{\lambda_{j}}} $ condition at all levels $ c_{\lambda_{j}} < \tau\lambda^{1- \frac{N}{ps}} $. By the usual critical point theory, all $ c_{\lambda_{j}} $ are critical levels and $ \Phi_{\lambda} $ has at least $ m $ pairs of nontrivial critical points satisfying

Thus, (2.1) has at least $ m $ pairs of solutions. Finally, as in the proof of Theorem 2.1, we know that these solutions satisfy the estimates (2.2) and (2.3).

5.   Conclusions

In this paper, we have obtained the existence and multiplicity of standing wave solutions for a class of perturbed fractional p-Laplacian systems involving critical exponents by variational methods. In the next work, we will extend the study to the case of perturbed fractional p-Laplacian systems with electromagnetic fields.

Acknowledgments

The author is grateful to the referees and the editor for their valuable comments and suggestions.

Conflict of interest

The author declares no conflict of interest.