On a class of three coupled fractional Schrödinger systems with general nonlinearities

1.   Introduction

In this paper, we study the following three-component coupled fractional Schrödinger system:

where $\alpha\in(0, 1)$, $d > 2\alpha$, $\omega_{j} > 0, j = 1, 2, 3$, $\lambda > 0$ is a coupling parameter and the fractional Laplacian $(-\Delta)^{\alpha}$ is given by

where

is a normalization constant, P.V. is the Cauchy principal value. We are dedicated to the existence of least energy solutions for system (1.1).

The fractional Laplacian operator $(-\Delta)^{\alpha}$ arises in several physical phenomena like fractional quantum mechanics and flames propagation, in population dynamics and geophysical fluid dynamics. In addition, $(-\Delta)^{\alpha}$ also arises in modeling diffusion and transport in a highly heterogeneous medium, or is used as an effective diffusion in a limiting advection-diffusion equation with a random velocity field. In ^[9], Laskin introduced the fractional Laplacian equation by expanding the Feynman path integral from the Brownian-like to the Lévy-like quantum mechanical paths. For more motivations and backgrounds, we refer the interested readers to ^[9,14] and references therein. From the mathematicians point of view, one of the main difficulties lies in that the fractional Laplacian $(-\Delta)^{\alpha}$ is a nonlocal operator.

Over the past few years, the following fractional Schrödinger equation has drawn many researchers' a great deal of attention

where $\alpha\in(0, 1)$. Equation (1.2) arises in looking for standing wave solutions for the fractional Schrödinger equation

where $i$ and $\Psi$ denote the imaginary unit and the wave function, respectively. Different approaches have been applied to deal with problem (1.2) under various hypotheses on the nonlinearity $f$, and several existence and nonexistence results via variational methods are obtained, see for example, ^[1,2,3] and the references therein. Recently, Guo and He ^[6] considered the system

where $\alpha\in(0, 1)$, $\omega > 0, b > 0$, $2q+2\in(2, 2_{\alpha}^{*})$. They, with the application of Nehari manifold method, proved (1.4) has a least energy solution. They also proved that if $b$ is large enough, system (1.4) has a positive least energy solution with both nontrivial components by using the similar arguments as in ^[13]. In ^[12], Lü and Peng investigated the following two-component coupled fractional Schrödinger system

Under some suitable assumptions on the nonlinear terms $f$ and $g$, they obtained the existence of positive solutions with both nontrivial components and least energy solutions with both nontrivial components for (1.5) by using variational methods. They also proved the asymptotic behavior of these solutions as the coupling parameter $\beta\rightarrow0$. More results concerning the fractional Schrödinger systems (1.4) and (1.5), can be seen in ^{[5,8,17,18,19]}.

To our best knowledge, there is no result in the literature on the existence result for three coupled fractional Schrödinger systems with general nonlinearities. We will prove some existence results for system (1.1). On a broader scale, this paper presumes that $f_{j}(j = 1, 2, 3)$ meet the conditions below:

$(A_{1})$ $f_{j}\in C^{1}(\mathbb{R}, \mathbb{R})$ and $f_{j}(t) = o(t)(t\rightarrow0^{+})$.

$(A_{2})$ There exist $q_{j}\in(2, 2_{\alpha}^{*})$ such that $\lim\limits_{|t|\rightarrow +\infty}\frac{f_{j}(t)}{|t|^{q_{j}-1}} = 0$, where $2_{\alpha}^{*} = \frac{2d}{d-2\alpha}$ is the fractional critical exponent.

$(A_{3})$ There exist $T_{j} > 0$ such that $F_{j}(T_{j}) > \frac{\omega_{j}}{2}T_{j}^{2}$, where $F_{j}(t): = \int_{0}^{t}f_{j}(s) {\mathrm{d}}s$.

When $\lambda = 0$, system (1.1) is converted to three uncoupled equations

In ^[2,3], it is proved that if $f_{j}$ satisfy $(A_{1})$–$(A_{3})$, (1.6) possesses a positive least energy solution $u_{j}^{\star}$ for $j = 1, 2, 3$. Hence, for all $\lambda\in \mathbb{R}$, the pairs $(u_{1}^{\star}, 0, 0)$, $(0, u_{2}^{\star}, 0)$ and $(0, 0, u_{3}^{\star})$ solve system (1.1). In this paper this sort of solutions (i.e., solutions with at least one trivial component) is called as semi-trivial solutions. An appealing question is whether the system (1.1) contains solutions $(u_{1}, u_{2}, u_{3})$ such that $u_{1}, u_{2}, u_{3}\not\equiv 0$ under the conditions $(A_{1})$–$(A_{3})$, such kind of solutions will be called fully nontrivial solutions. The major results are the following:

Theorem 1.1. Suppose that $f_{j}$ satisfy $(A_{1})-(A_{3})$ for $j = 1, 2, 3$ and $2\alpha < d < 6\alpha$. Then

(i) for any $\lambda > 0$, the system (1.1) has a least energy solution,

(ii) there exists $\lambda^{*} > 0$ such that for every $\lambda > \lambda^{*}$, the system (1.1) has a fully nontrivial least energy solution.

Remark 1.1. Theorem 1.1 can be thought of as an extension of the results in ^[3,6,16]. We note that in our assumptions $(A_{1})$–$(A_{3})$ neither any monotonicity condition nor any Ambrosetti-Rabinowitz growth condition is required, and we need a new method different from those used in ^[3,6,16].

The structure of the other parts of the paper is as follows. In Section 2, some notations and preliminary results are proposed. In Section 3, we conclude the proof of Theorem 1.1.

2.   Preliminaries

Throughout this paper, $C, C_{i}$ will signify different kinds of positive constants; the strong convergence is denoted by $\rightarrow$, and the weak convergence denoted by $\rightharpoonup$; $B_{\rho}(y)$ denotes a ball centered at $y$ with radius $\rho > 0$; $\|u\|_{L^{q}(\mathbb{R}^{d})} = (\int_{\mathbb{R}^{d}}|u|^{q}{\mathrm{d}}x)^{\frac{1}{q}}$ denote the norm of $L^{q}(\mathbb{R}^{d})$. The fractional Sobolev space $H^{\alpha}(\mathbb{R}^{d})$ is marked as

endowed with the norm

where

is the so-called Gagliardo semi-norm of $w$. Via Fourier transform, we have

where the symbol $\ \widehat{} \ $ stands for Fourier transform. Therefore, by the Fourier transform, $H^{\alpha}(\mathbb{R}^{d})$ can be equivalently defined as follows

and the norm can be equivalently written

From Propositions 3.4 and 3.6 in ^[14], for any $w\in H^{\alpha}(\mathbb{R}^{d})$, we have

We also use the following notations:

$(1)$ $\mathcal{D}^{\alpha, 2}(\mathbb{R}^{d})$ is completion of $C_{0}^{\infty}(\mathbb{R}^{d})$ concerning the norm

$(2)$ For $\omega_{j} > 0, j = 1, 2, 3$, we use the notation

which is an equivalent norm to $\|w\|_{H^{\alpha}(\mathbb{R}^{d})}$.

$(3)$

with the norm

and

where

For the fractional Sobolev spaces, the embedding results below can be got in ^[14].

Lemma 2.1. If $\alpha\in(0, 1)$ and $d > 2\alpha$, then

(i) $\mathcal{D}^{\alpha, 2}(\mathbb{R}^{d})$ is continuously embedded into $L^{2_{\alpha}^{*}}(\mathbb{R}^{d})$, i.e.,

for any $w\in\mathcal{D}^{\alpha, 2}(\mathbb{R}^{d})$, where constant $C$ depending only on $d, \alpha$.

(ii) $H^{\alpha}(\mathbb{R}^{d})\hookrightarrow L^{q}(\mathbb{R}^{d})$ is continuous for any $q\in[2, 2_{\alpha}^{*}]$.

(iii) $H^{\alpha}(\mathbb{R}^{d})\hookrightarrow L_{loc}^{q}(\mathbb{R}^{d})$ is compact for any $q\in[1, 2_{\alpha}^{*})$; $H_r^{\alpha}(\mathbb{R}^{d})\hookrightarrow L^{q}(\mathbb{R}^{d})$ is compact for any $q\in(2, 2_{\alpha}^{*})$.

3.   Proof of Theorem 1.1

This section is devoted to proving the Theorem 1.1. Define a functional related to system (1.1) by

where for $j = 1, 2, 3$,

On the basis of the conditions $(A_{1})$–$(A_{3})$, one can easily verify that $\Phi_{\lambda}$ is well defined and $C^{1}$. Now, we define the Pohozaev set by

where

From $(A_{1})$–$(A_{3})$, if $(u_{1}, u_{2}, u_{3})\in \mathscr{H}$ is a weak solution to system (1.1), using the similar regularity arguments as in ^[2], we can get $u_{j}\in C^{1}(\mathbb{R}^{d})$ for $j = 1, 2, 3$. Then it is classical to confirm that each nontrivial solution of (1.1) belongs to $\mathscr{N}_{\lambda}$. Moreover, we have

Lemma 3.1. Let the conditions $(A_{1})$–$(A_{3})$ hold, then

(i) $\mathscr{N}_{\lambda}$ is a $C^{1}$ manifold,

(ii) for any $(u_{1}, u_{2}, u_{3})\in\mathscr{N}_{\lambda}$, there exists constant $\varrho_{0} > 0$ such that $\|(u_{1}, u_{2}, u_{3})\|\geq \varrho_{0}$,

(iii) if $u_{i}\in H^{\alpha}(\mathbb{R}^{d})\setminus\{0\}$ and $\mathcal{N}(u_{1}, u_{2}, u_{3})\leq0$, then exists a unique $\bar{t}\in(0, 1]$ such that

where $u_{i}^{t}(x) = u_{i}(t^{-1}x)$.

Proof. (i) From $(A_{1})$–$(A_{3})$, we know that $\mathcal{N}(u_{1}, u_{2}, u_{3})$ is a $C^{1}$ functional, in order to prove $\mathscr{N}_{\lambda}$ is a $C^{1}$ manifold, it suffices to prove that $\mathcal{N}'(u_{1}, u_{2}, u_{3})\neq0$ for all $(u_{1}, u_{2}, u_{3})\in \mathscr{N}_{\lambda}$. Indeed, assume by contradiction that $\mathcal{N}'(u_{1}, u_{2}, u_{3}) = 0$ for some $(u_{1}, u_{2}, u_{3})\in \mathscr{N}_{\lambda}$. Then in a weak sense, $(u_{1}, u_{2}, u_{3})$ can be seen as a solution of the system

As a consequence, we see that $(u_{1}, u_{2}, u_{3})$ satisfies the Pohozaev type identity referred to (3.2), that is

Since $\mathcal{N}(u_{1}, u_{2}, u_{3}) = 0$, by (3.3) we deduce that

which implies that $u_{j} = 0$ for all $j = 1, 2, 3$, which is a contradiction since $(u_{1}, u_{2}, u_{3})\in \mathscr{N}_{\lambda}$. Thus $\mathscr{N}_{\lambda}$ is a $C^{1}$ manifold.

(ii) Let $(u_{1}, u_{2}, u_{3})\in\mathscr{N}_{\lambda}$, then we have

From the conditions $(A_{1})$ and $(A_{2})$, we know that, for any $\epsilon > 0$, there is $C_{\epsilon} > 0$ such that

Then by (3.4), (3.5) and the Sobolev embedding inequality, one has

which implies that $\|(u_{1}, u_{2}, u_{3})\|\geq \varrho_{0}$ for some positive constant $\varrho_{0} > 0$ since $q_{1}, q_{2}, q_{3} > 2$.

(iii) Let $u_{1}, u_{2}, u_{3}\in H^{\alpha}(\mathbb{R}^{d})\setminus \{0\}$ satisfy $\mathcal{N}(u_{1}, u_{2}, u_{3})\leq0$, then

For $t > 0$, let $u_{j}^{t} = u_{j}(t^{-1}x), j = 1, 2, 3$, we define

Obviously, $h(t)\rightarrow -\infty$ as $t\rightarrow +\infty$. Moreover, $h(t) > 0$ for $t > 0$ small enough. In fact, from $(A_{1})$ and $(A_{2})$, for any $\epsilon > 0$, there is $C_{\epsilon} > 0$ such that for all $\tau\in \mathbb{R}$,

Then by (3.7), Hölder inequality and Lemma 2.1(i), we get

which yields that $h(t) > 0$ when $t > 0$ sufficiently small. Hence, we can find $\bar{t} > 0$ such that $h(t)$ has a positive maximum and $h'(\bar{t}) = 0$. Notice that

so we get $\mathcal{N}(u_{1}^{\bar{t}}, u_{2}^{\bar{t}}, u_{3}^{\bar{t}}) = 0$. Furthermore, by $\mathcal{N}(u_{1}^{\bar{t}}, u_{2}^{\bar{t}}, u_{3}^{\bar{t}}) = 0$, we can deduce that

By (3.6), we know that $\bar{t}\in(0, 1]$. Hence, $\bar{t}\in(0, 1]$ is the unique critical point of $h(t)$ corresponding to its maximum. □

Let us define the least energy

We call that a minimizer on $\mathscr{N}_{\lambda}$ is a least energy solution for system (1.1). By the proof of Lemma 3.1(ii) and (iii), it is clear that the functional $\Phi_{\lambda}$ satisfies the mountain-pass geometry. Let $c^{*}$ be the minmax mountain-pass level for the functional $\Phi_{\lambda}$ given by

where

Arguing as in the proof of Lemma 4.2 in ^[10], we can get that $c_{\lambda} = c^{*} > 0$.

Lemma 3.2. $(u_{1}, u_{2}, u_{3})$ is a solution of system (1.1) provided $c_{\lambda}$ is attained at $(u_{1}, u_{2}, u_{3})\in \mathscr{N}_{\lambda}$, where $c_{\lambda}$ is defined in (3.8).

Proof. Suppose that $(u_{1}, u_{2}, u_{3})\in \mathscr{N}_{\lambda}$ such that $\Phi_{\lambda}(u_{1}, u_{2}, u_{3}) = c_{\lambda}$. Then by the theory of Lagrange multipliers, there exists a Lagrange multiplier $\mu\in \mathbb{R}$ such that

As a consequence, $(u_{1}, u_{2}, u_{3})$ satisfies the following Pohozaev type identity

Notice that $\mathcal{N}(u_{1}, u_{2}, u_{3}) = 0$, from (3.4) and (3.9) we obtain

which implies that $\mu = 0$. Thus $\Phi'_{\lambda}(u_{1}, u_{2}, u_{3}) = 0$, and so $(u_{1}, u_{2}, u_{3})$ is a solution of system (1.1). □

Lemma 3.3. Let $\alpha\in(0, 1)$ and $d > 2\alpha$. Suppose that $\{w_{n}\}$ is a bounded sequence in $H^{\alpha}(\mathbb{R}^{d})$ and

Then $w_{n}\rightarrow0$ in $L^{q}(\mathbb{R}^{d})$ for every $q\in(2, 2_{\alpha}^{*})$.

The above vanishing lemma has been proved in ^[4] (see Lemma 2.2). Then we have:

Lemma 3.4. If $\{(u_{1, n}, u_{2, n}, u_{3, n})\}\subset\mathscr{N}_{\lambda}$ is a bounded sequence, there exists a sequence $\{z_n\} \subset \mathbb{R}^{d}$ and constants $R, \eta > 0$ such that

Proof. Arguing from the reversed point, suppose that the conclusion does not hold, then for every $R > 0$, one has

By (3.10) and Lemma 3.3, we obtain that for all $r\in (2, 2_{\alpha}^{*})$, $u_{j, n}\rightarrow 0\ \ \mbox{in} \ L^{r}(\mathbb{R}^{d})$ for $j = 1, 2, 3$. Furthermore, notice that $\{(u_{1, n}, u_{2, n}, u_{3, n})\}\subset\mathscr{N}_{\lambda}$, then we can deduce that $(u_{1, n}, u_{2, n}, u_{3, n})\rightarrow (0, 0, 0)$ in $\mathscr{H}$. On the other hand, from Lemma 3.1(ii), we have $\|(u_{1, n}, u_{2, n}, u_{3, n})\|\geq \varrho_{0}$ for some $\varrho_{0} > 0$, so we get a contradiction, the proof is finished. □

Lemma 3.5. Suppose that $f_{j}$ satisfy $(A_{1})$–$(A_{3})$ for $j = 1, 2, 3$ and $2\alpha < d < 6\alpha$. Then for each $\lambda > 0$, there exists

such that

Proof. Suppose $\{(u_{1, n}, u_{2, n}, u_{3, n})\}\subset \mathscr{N}_{\lambda}$ such that

Let $(u_{1, n}^{*}, u_{2, n}^{*}, u_{3, n}^{*})$ be the Schwarz symmetrization of $(u_{1, n}, u_{2, n}, u_{3, n})$, by the fractional Polya-Szegö inequality (see Theorem 2.1 in ^[7] or Theorem 1.1 in ^[15]) and the properties of the Schwarz symmetrization (see Lieb-Loss ^[11]) and Lemma 3.1(iii), there exists $\tilde{t}_{n}\in(0, 1]$ such that

and

Hence we can assume that $u_{1, n}, u_{2, n}$ and $u_{3, n}$ are radial, i.e.,

First, we note that $\{(u_{1, n}, u_{2, n}, u_{3, n})\}$ is bounded in $\mathscr{H}_r$. Indeed, since $\{(u_{1, n}, u_{2, n}, u_{3, n})\}\subset \mathscr{N}_{\lambda}$, we have $\mathcal{N}(u_{1, n}, u_{2, n}, u_{3, n}) = 0$, then we infer that

By (3.11), we get that $\{u_{j, n}\}$ are bounded in $\mathcal{D}^{\alpha, 2}(\mathbb{R}^{d})$ for all $j = 1, 2, 3$. On the other hand, since $\{(u_{1, n}, u_{2, n}, u_{3, n})\}\subset \mathscr{N}_{\lambda}$ and note that $2_{\alpha}^{*} > 3$, then by (3.4), (3.7), Young inequality and Lemma 2.1, we can deduce that $\|u_{j, n}\|_{L^{2}(\mathbb{R}^{d})}$ are bounded for $j = 1, 2, 3$. Therefore, $\{(u_{1, n}, u_{2, n}, u_{3, n})\}$ is bounded in $\mathscr{H}_r$, and then there exist $u_{1}, u_{2}, u_{3}\in H^{\alpha}_r (\mathbb{R}^{d})$ such that for $j = 1, 2, 3$,

From Lemma 3.4, we know that there exists $\{z_n\} \subset \mathbb{R}^{d}$ and constants $R, \eta > 0$ satisfying

Now we define

from the invariance of $\mathbb{R}^{d}$ by translations, then $\{(\widetilde{u}_{1, n}(x), \widetilde{u}_{2, n}(x), \widetilde{u}_{3, n}(x))\}$ is also a minimizing sequence for $c_{\lambda}$. Hence, by arguing as we did for $\{(u_{1, n}, u_{2, n}, u_{3, n})\}$, passing to a subsequence, we can assume that

in $\mathscr{H}_r$, $(\widetilde{u}_{1, n}, \widetilde{u}_{2, n}, \widetilde{u}_{3, n})\rightarrow (\tilde{u_{1}}, \tilde{u_{2}}, \tilde{u_{3}})$ in $L_{loc}^{2}(\mathbb{R}^{d})\times L_{loc}^{2}(\mathbb{R}^{d})\times L_{loc}^{2}(\mathbb{R}^{d})$. Additionally, by (3.12),

From (3.13), one has

and so $(\tilde{u_{1}}, \tilde{u_{2}}, \tilde{u_{3}})\neq(0, 0, 0)$. On the other hand, since

passing to the limit, we get

then by Lemma 3.1(iii) there is $\tilde{t}\in(0, 1]$ such that

Thus we have

hence $\Phi_{\lambda}(\widetilde{u_{1}}_{\lambda}, \widetilde{u_{2}}_{\lambda}, \widetilde{u_{3}}_{\lambda}) = c_{\lambda}$ and $(\widetilde{u_{1}}_{\lambda}, \widetilde{u_{2}}_{\lambda}, \widetilde{u_{3}}_{\lambda})$ is a minimizer of $\Phi_{\lambda}$ restricted to $\mathscr{N}_{\lambda}$. □

Now the proof of Theorem 1.1 will be presented.

Proof of Theorem 1.1. (i) From Lemma 3.5, we know that there exists

such that $\Phi_{\lambda}(\widetilde{u_{1}}_{\lambda}, \widetilde{u_{2}}_{\lambda}, \widetilde{u_{3}}_{\lambda}) = c_{\lambda}.$ Then by Lemma 3.2, we have that $\Phi'_{\lambda}(\widetilde{u_{1}}_{\lambda}, \widetilde{u_{2}}_{\lambda}, \widetilde{u_{3}}_{\lambda}) = 0$, that is, $(\widetilde{u_{1}}_{\lambda}, \widetilde{u_{2}}_{\lambda}, \widetilde{u_{3}}_{\lambda})$ is a least energy solution for the system (1.1). The Theorem 1.1(i) is proved.

(ii) For $j = 1, 2, 3$, let $u_{j}^{\star}\in H^{\alpha}_r(\mathbb{R}^{d})$ be the positive least energy solutions respectively for Eq (1.6). Then it is easy to see that $\mathcal{N}(u_{1}^{\star}, u_{2}^{\star}, u_{3}^{\star})\leq0$, from Lemma 3.1(iii), there is $t^{*} > 0$ such that

Notice that the pairs $(u_{1}^{\star}, 0, 0)$, $(0, u_{2}^{\star}, 0)$ and $(0, 0, u_{3}^{\star})$ solve system (1.1), and system (1.1) has no solutions with exactly one trivial component. Now, on the basis of idea from ^[13] (or see ^[6]), to indicate the radial least energy solution of system (1.1) is a fully nontrivial least energy solution, we just need to prove that, for $\lambda > 0$ sufficiently large,

Indeed, by some calculations, we can infer that

Therefore, when $\lambda > 0$ large enough, (3.14) holds. Thus the Theorem 1.1(ii) follows. □

4.   Conclusions

In this paper, we are interested in studying a class of systems of three-component coupled nonlinear fractional Schrödinger equations with general nonlinearities. In our assumptions $(A_{1})$–$(A_{3})$ neither any monotonicity condition nor any Ambrosetti-Rabinowitz growth condition is required, so we need to overcome several difficulties when using variational methods. By using a Pohozaev manifold method and variational arguments, we establish some novel existence results of least energy solutions for the three-component coupled fractional Schrödinger system (1.1). We believe that the proposed approach in this paper can also be applied to study other related equations and systems.

Acknowledgments

The authors are grateful to the reviewers and editors for their valuable comments and suggestions for improvement of the paper. This work is partially supported by the fund from NSFC (12126423) and the Research Project of Hubei Engineering University (202231).

Conflict of interest

The authors declare that they have no competing interests.