Ground states to a Kirchhoff equation with fractional Laplacian

1.   Introduction

We are concerned with the existence and decay property of ground state solutions for the following Kirchhoff equation involving fractional Laplacian operator:

where $\alpha, \beta, \kappa > 0$ are positive constants, $s\in(0, 1)$, $2 < p < 2_{s}^{*} = \frac{6}{3-2s}$ and the fractional Laplacian $(-\Delta)^{s}$ is given by

where

The fractional Laplacian operator $(-\Delta)^{s}$ has a wide range of applications arising in some physical phenomena such as fractional quantum mechanics, flames propagation, etc. (see ^[10,13]). In recent years, problems involving fractional Laplacian operators and Kirchhoff-type nonlocal terms have been discussed by lots of researchers for their broad applications. Some remarkable results have been yielded, see ^{[1,2,6,7,8,12,14,15]} and the references therein. In particular, when $\alpha = 1$, $\beta = 0$ and $\mathbb{R}^{3}$ is replaced by $\mathbb{R}^{N}(N\geq2)$, equation (1.1) turns into the classical fractional Laplacian problem

In ^[4], employing the constrained variational methods, Dipierro et al. studied the existence and symmetry of nontrivial solutions for (1.2) as $s\in(0, 1)$ and $p\in (2, \frac{2N}{N-2s})$.

In this paper, we intend to consider the fractional Kirchhoff equation (1.1) with $p\in (2, 2_{s}^{*})$ using variational arguments, and we encounter several difficulties to overcome. First, note that solutions of Eq (1.1) correspond to critical points of the following functional:

Since the nonlocal term $\big(\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}} \psi|^{2}\, \mathrm{d} x\big)^{2}$ included in the energy functional $\mathcal{E}(\psi)$ is homogeneous of degree $4$, and the nonlinearity $|\psi|^{p-2}\psi$ does not satisfy the global Ambrosetti-Rabinowitz type condition for $p\in (2, 2_{s}^{*})$, it would bring about more difficulties to establish the boundedness of (PS)-sequence for $\mathcal{E}(\psi)$ when $p\leq4$. Second, in general, from $\psi_{n}\rightharpoonup \psi$ in $H^{s}(\mathbb{R}^{3})$, we do not know whether there holds

which is vital when we consider the convergence of the (PS)-sequence.

We now give the main result.

Theorem 1.1. Let $s\in(\frac{3}{4}, 1)$ and $p\in (2, 2_{s}^{*})$. Then, Eq (1.1) has a ground state solution $\psi_{0}(x)\in H^{s}(\mathbb{R}^{3})$, namely

Moreover, $\psi_{0}(x)\leq \frac{C}{1+|x|^{3+2s}}$ for some constant $C > 0$.

Remark 1.1. In Theorem 1.1, we give the existence result for all $p\in (2, 2_{s}^{*})$, our result could be viewed as an extension of one of the main results in ^[11] (see in particular Theorem 1.4 there), which only dealt with the case $p\in (3, 2_{s}^{*})$ with $s = 1$, the case $p\in (2, 3]$ having been left open.

2.   Proof of Theorem 1.1

In this paper, we use the notation $\|\psi\|_{L^{q}(\mathbb{R}^{3})} = (\int_{\mathbb{R}^{3}}|\psi|^{q}\, \mathrm{d} x)^{\frac{1}{q}}$ to denote the norm of $L^{q}(\mathbb{R}^{3}), q\in[1, +\infty)$. For $s\in(0, 1)$, the fractional Sobolev space $H^{s}(\mathbb{R}^{3})$ is defined as

endowed with the norm

Note that by Propositions 3.4 and 3.6 in ^[13], one has

and so $\|\psi\|_{H^{s}(\mathbb{R}^{3})} = (\int_{\mathbb{R}^{3}}(|(-\Delta)^{\frac{s}{2}} \psi|^{2}+|\psi|^{2})\, \mathrm{d} x)^{\frac{1}{2}}$ is an equivalent norm to $\|\psi\|_{H^{s}}$ for $\psi\in H^{s}(\mathbb{R}^{3})$.

For fixed $\alpha, \kappa > 0$, we also employ the norm $\|\psi\| = (\int_{\mathbb{R}^{3}}(\alpha|(-\Delta)^{\frac{s}{2}} \psi|^{2}+\kappa|\psi|^{2})\, \mathrm{d} x)^{\frac{1}{2}}$, which is an equivalent norm to $\|\psi\|_{H^{s}(\mathbb{R}^{3})}$. $\mathcal{D}^{s, 2}(\mathbb{R}^{3})$ is the completion of $C_{0}^{\infty}(\mathbb{R}^{3})$ with respect to the norm $\|\psi\|_{\mathcal{D}^{s, 2}(\mathbb{R}^{3})} = (\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}} \psi|^{2}\, \mathrm{d}x)^{\frac{1}{2}}$. Now, we recall the following fractional Sobolev embedding results.

Lemma 2.1. (See ^[13]) Let $s\in(0, 1)$. Then, the embeddings $\mathcal{D}^{s, 2}(\mathbb{R}^{3})\hookrightarrow L^{2_{s}^{*}}(\mathbb{R}^{3})$ and $H^{s}(\mathbb{R}^{3})\hookrightarrow L^{q}(\mathbb{R}^{3})(q\in[2, 2_{s}^{*}])$ are continuous, and the embedding $H^{s}(\mathbb{R}^{3})\hookrightarrow L_{loc}^{q}(\mathbb{R}^{3})(q\in[1, 2_{s}^{*}))$ is compact.

We define the minimax value

where

First, we show that $\Lambda\neq\varnothing$, it is sufficient to prove the following lemma.

Lemma 2.2. Let $s\in(\frac{3}{4}, 1)$ and $p\in (2, 2_{s}^{*})$. Then, there exists $\psi_{*}\in H^{s}({\mathbb{R}}^{3})$ such that $\mathcal{E}(\psi_{*}) < 0$.

Proof. We consider the following perturbation functional $\mathcal{E}_{\eta}$ defined by

where $G_{\eta}(\psi): = \frac{\eta}{p}|\psi|^{p}-\frac{\kappa}{2}|\psi|^{2}$ and $\eta\in[\eta_{0}, 1]$ is a parameter, $\eta_{0}\in(0, 1)$ is a positive constant. Now, we take $t_{0} > 0$ such that

and for $\ell > 0$ define

By the definition of $\varphi_{\ell}(x)$, clearly $\varphi_{\ell}(x)\in H^{s}({\mathbb{R}}^{3})$ and $\|\varphi_{\ell}\|_{H^{s}({\mathbb{R}}^{3})}\rightarrow +\infty$ as $\ell\rightarrow +\infty$. Moreover, by direct calculations (see Lemma 2.6 in ^[4]), we conclude that

and

where $C_{0}, C_{1}, C_{2}$ are constants depending on $t_{0}$. Thus, for sufficiently large $\ell_{0} > 0$ we can get

Let $\varphi_{\ell_{0}, \sigma}(x) = \varphi_{\ell_{0}}(\frac{x}{\sigma})$ for $\sigma > 0$, then the following hold true:

Hence, by (2.2) and note that $6-4s < 3$ we have

In addition, we notice that $\mathcal{E}(\psi)\leq \mathcal{E}_{\eta_{0}}(\psi)$ for any $\psi\in H^{s}({\mathbb{R}}^{3})$, so we obtain $\mathcal{E}(\varphi_{\ell_{0}, \sigma})\rightarrow -\infty$ as $\sigma\rightarrow +\infty$. Thus, we can take $\psi_{*} = \varphi_{\ell_{0}, \sigma_{0}}$ such that $\mathcal{E}(\psi_{*}) < 0$ for $\sigma_{0} > 0$ large enough.

By Lemma 2.1, for all $\psi\in H^{s}(\mathbb{R}^{3})$, we know that $\|\psi\|_{L^{p}(\mathbb{R}^{3})}^{p}\leq c\|\psi\|^{p}$ for some positive constant $c > 0$, and noting that $p > 2$, we deduce that

if $\|\psi\| = \varepsilon_{0} > 0$ is sufficiently small. Thus, combining with Lemma 2.2, we know that $c_{mp}\in(0, +\infty)$. Note that if $\psi\in H^{s}({\mathbb{R}}^{3})$ is a critical point of $\mathcal{E}$, then $\psi$ satisfies the Pohozaev identity (see Lemma 2.2 in ^[15]):

Next, we expound that there is a Palais-Smale-Pohozaev sequence ((PSP)-sequence, for short) at the minimax level $c_{mp}$ defined by (2.1).

Lemma 2.3. Let $s\in(\frac{3}{4}, 1)$, $p\in (2, 2_{s}^{*})$ and $\kappa > 0$. Then, there is a sequence $\{\psi_{n}\}\subset H^{s}({\mathbb{R}}^{3})$ such that

(ⅰ) $\mathcal{E}(\psi_{n})\rightarrow c_{mp}$ as $n\rightarrow \infty$;

(ⅱ) $\mathcal{E}'(\psi_{n})\rightarrow 0$ in $(H^{s}({\mathbb{R}}^{3}))^{*}$ as $n\rightarrow \infty$;

(ⅲ) $\mathcal{P}(\psi_{n})\rightarrow 0$ as $n\rightarrow \infty$.

Proof. For $\tau\in \mathbb{R}$ and $\varphi\in H^{s}({\mathbb{R}}^{3})$, we set $(\bf{S}_{\tau}\varphi)(x) = \varphi(e^{-\tau}x)$, and we denote $\widetilde{\mathcal{E}}: \mathbb{R}\times H^{s}({\mathbb{R}}^{3})\rightarrow \mathbb{R}$ the functional defined by

Here, $\mathbb{R}\times H^{s}({\mathbb{R}}^{3})$ is equipped with the norm $\|(\tau, \varphi)\|_{\mathbb{R}\times H^{s}({\mathbb{R}}^{3})} = (|\tau|^{2}+\|\varphi\|^{2})^{\frac{1}{2}}$. Clearly, $\widetilde{\mathcal{E}}(\tau, \varphi)\in \mathcal{C}^{1}(\mathbb{R}\times H^{s}({\mathbb{R}}^{3}))$ and satisfies the following properties:

We define a minimax value for $\widetilde{\mathcal{E}}(\tau, \varphi)$ by

where

It is not hard to see that $\widetilde{\Lambda}\neq \emptyset$, and thus the minimax value $d_{mp}$ is well defined. Now, we claim that $d_{mp} = c_{mp}$. Indeed, for any $\zeta(t)\in \Lambda$ we can check that $(0, \zeta(t))\in \widetilde{\Lambda}$, so $\{0\}\times\Lambda\subset \widetilde{\Lambda}$, thus for $\widetilde{\mathcal{E}}(0, \varphi) = \mathcal{E}(\varphi)$ we get $d_{mp}\leq c_{mp}$. On the other hand, for every given $\widetilde{\zeta}(t) = (\tau(t), \eta(t))\in \widetilde{\Lambda}$, letting $\zeta(t)(x) = \eta(t)(e^{-\tau(t)}(x))$, we can verify that $\zeta(t)\in \Lambda$, and from (2.4), we obtain $\mathcal{E}(\zeta(t)) = \widetilde{\mathcal{E}}(\widetilde{\zeta}(t))$. This yields that $d_{mp}\geq c_{mp}$. Hence, the claim follows.

By (2.1), we may choose $\{\zeta_{n}\}\subset \Lambda$ such that

Let $\tilde{\zeta}_{n}(t): = (0, \zeta_{n}(t))$, then $\tilde{\zeta}_{n}\in\widetilde{\Lambda}$ and so, we obtain

Then, by Lemma 2.3 in ^[9], we can get that a sequence $\{(\tau_{n}, \varphi_{n})\}\subset\mathbb{R}\times H^{s}({\mathbb{R}}^{3})$ satisfies

and

Remark that for any $(h, U)\in \mathbb{R}\times H^{s}({\mathbb{R}}^{3})$,

Then, the conclusion of Lemma 2.3 follows by taking $\psi_{n} = \bf{S}_{\tau_{n}}\varphi_{n}$. Indeed, by (2.6) we can get

By the fact that $\tau_{n}\rightarrow 0$ as $n\rightarrow \infty$, via (2.5) and (2.4) one can obtain that $\mathcal{E}(\psi_{n})\rightarrow c_{mp}$ as $n\rightarrow \infty$.

For any $\varphi\in H^{s}({\mathbb{R}}^{3})$, we choose $h = 0, U(x) = \varphi(e^{\tau_{n}}x)$ in (2.7), and note that $\tau_{n}\rightarrow 0$, then we obtain

Hence, $\mathcal{E}'(\psi_{n})\rightarrow 0$ as $n\rightarrow \infty$. Moreover, taking $(h, U) = (1, 0)$ in (2.7), we get $\mathcal{P}(\psi_{n})\rightarrow 0$ as $n\rightarrow \infty$.

To sum up, we have obtained a sequence $\{\psi_{n}\}\subset H^{s}({\mathbb{R}}^{3})$ that satisfies

The proof is completed.

Lemma 2.4. The (PSP)-sequence $\{\psi_{n}\}$ in (2.8) is bounded in $H^{s}({\mathbb{R}}^{3})$.

Proof. From (2.8), we get

Note that by (2.9), $\|(-\Delta)^{\frac{s}{2}} \psi_{n}\|_{L^{2}(\mathbb{R}^{3})}$ is bounded. By Lemma 2.1, the fractional Sobolev embedding $D^{s, 2}({\mathbb{R}}^{3})\hookrightarrow L^{2_{s}^{*}}({\mathbb{R}}^{3})$ is continuous, so we have

and thus $\|\psi_{n}\|_{L^{2_{s}^{*}}(\mathbb{R}^{3})}$ is bounded. Next, we prove $\{ \psi_{n}\}$ is bounded in $L^{2}(\mathbb{R}^{3})$. By the fact that $\mathcal{E}'(\psi_{n})\rightarrow 0$ and $\|(-\Delta)^{\frac{s}{2}} \psi_{n}\|_{L^{2}(\mathbb{R}^{3})}$ is bounded, we can deduce that

for some constant $C > 0$. Since $2 < p < 2_{s}^{*}$, then for any $\varepsilon > 0$, there is $C_{\varepsilon} > 0$ such that

Thus, by (2.10) we obtain

Choosing $\varepsilon = \frac{\kappa}{2}$, and meanwhile $\|\psi_{n}\|_{L^{2_{s}^{*}}(\mathbb{R}^{3})}$ is bounded, we obtain the boundedness of $\|\psi_{n}\|_{L^{2}(\mathbb{R}^{3})}$ and therefore $\{\psi_{n}\}$ is bounded in $H^{s}({\mathbb{R}}^{3})$.

Proof of Theorem 1.1. The result of Lemma 2.3 reveals that there is a (PSP)-sequence $\{\psi_{n}\}\subset H^{s}({\mathbb{R}}^{3})$ satisfying

Moreover, by Lemma 2.4 the (PSP)-sequence $\{\psi_{n}\}$ must be bounded in $H^{s}({\mathbb{R}}^{3})$. Then, passing to a subsequence if necessary, we may suppose that

Next, we divide our arguments into several steps.

Step 1: We claim that $\psi_{0}$ solves Eq $(1.1)$. In fact, by (2.11) for any $\xi\in C_{0}^{\infty}(\mathbb{R}^{3})$, we have

and

Moreover, suppose that $\|(-\Delta)^{\frac{s}{2}} \psi_{n}\|_{L^{2}(\mathbb{R}^{3})}^{2}\rightarrow \mathscr{B}$ for some $\mathscr{B}\geq0$, then from (2.12) and (2.13), for $\xi\in C_{0}^{\infty}(\mathbb{R}^{3})$, we deduce that

where

Using Fatou's lemma, we have

Noting that $\mathcal{E}'(\psi_{n})\rightarrow0$ and $\|(-\Delta)^{\frac{s}{2}} \psi_{n}\|_{L^{2}(\mathbb{R}^{3})}^{2}\rightarrow \mathscr{B}$, one can obtain that

Now, by combining (2.13)–(2.15) we conclude that

Putting together (2.16) and (2.15), we get

Accordingly, using (2.12), (2.13) and (2.17), we can derive that $\psi_{n}\rightarrow \psi_{0}$ strongly in $H^{s}({\mathbb{R}}^{3})$ and so

for all $\xi\in C_{0}^{\infty}(\mathbb{R}^{3})$, that is, $\mathcal{E}'(\psi_{0}) = 0$.

Moreover $\psi_{0}\not\equiv0$. Otherwise, if $\psi_{0}\equiv0$, that is $\psi_{n} \rightarrow0$ in $H^{s}({\mathbb{R}}^{3})$, which leads to $\mathcal{E}(\psi_{n})\rightarrow0$, this is a contradiction since $\mathcal{E}(\psi_{n})\rightarrow c_{mp} > 0$.

Step 2: Next, we claim that $\mathcal{E}(\psi_{0}) = c_{mp} = \mathfrak{m}$, that is, $\psi_{0}$ is a ground state solution of (1.1). Indeed, note that $\mathcal{P}(\psi_{n})\rightarrow0$ as $n\rightarrow \infty$, one has $\mathcal{P}(\psi_{0}) = 0$. Therefore, from (2.3),

Clearly, by the definition of $\mathfrak{m}$, there holds $\mathfrak{m}\leq \mathcal{E}(\psi_{0})$, and hence $\mathfrak{m}\leq c_{mp}$.

On the other hand, we prove that $c_{mp}\leq \mathfrak{m}$. Let $w(x)\in H^{s}({\mathbb{R}}^{3})\setminus\{0\}$ be another solution of $(1.1)$ and satisfy $\mathcal{E}(w)\leq \mathcal{E}(\psi_{0})$. We set $\zeta^{*}(\tau)(x) = w(\frac{x}{\tau})$ for $\tau > 0$ and $\zeta^{*}(0) = 0$. It is clear that $\zeta^{*}(\tau)\in \mathcal{C}([0, +\infty), H^{s}({\mathbb{R}}^{3}))$. From (2.3), for $\tau > 0$ we obtain that

With a simple calculation, we conclude that

and it follows that $\mathcal{E}(\zeta^{*}(\tau))\leq \mathcal{E}(w)$. Observe that $\mathcal{E}(\zeta^{*}(\tau))\rightarrow -\infty$ as $\tau\rightarrow +\infty$. Then, with appropriate scaling change we can get a path $\zeta(t)\in \mathcal{C}([0, 1], H^{s}({\mathbb{R}}^{3}))$ such that $\zeta(0) = 0$ and $\mathcal{E}(\zeta(1)) < 0$; $\zeta(t_{0}) = w$ for some $t_{0}\in(0, 1)$; $\max_{0\leq t\leq1}\mathcal{E}(\zeta(t)) = \mathcal{E}(\zeta(t_{0})) = \mathcal{E}(w)$. Then, by the definition of $c_{mp}$ in (2.1), we know that $c_{mp}\leq \mathcal{E}(w)$, which shows that $c_{mp}\leq \mathfrak{m}$. Thus as desired $\mathcal{E}(w) = \mathcal{E}(\psi_{0}) = c_{mp} = \mathfrak{m}$ has been proved.

Step 3: We estimate the decay properties of $\psi_{0}(x)$. Following ^[3], by the standard regularity arguments we can deduce that $\psi_{0}(x)\in H^{2s}(\mathbb{R}^{3})\cap C^{r}(\mathbb{R}^{3})$ for all $r\in(0, 2s)$ and $\lim\limits_{|x|\rightarrow \infty}\psi_{0}(x) = 0$. Note that $p > 2$. Then, we can pick $\rho > 0$ such that for all $|x|\geq \rho$,

where $L > 0$ such that $\|\psi_{0}\|_{H^{s}(\mathbb{R}^{3})}^{2}\leq L$, and we conclude that

Therefore,

According to Lemma 4.3 of ^[5], we can find a continuous function $\Phi(x)$ satisfying $0 < \Phi(x)\leq \frac{C}{1+|x|^{3+2s}}$ and

for some suitable $R_{1} > 0$. Let $R = \max\{\rho, R_{1}\}$, and set

Define $U(x) = \frac{b}{a}\Phi(x)-\psi_{0}(x)$. From (2.18) and (2.19), consequently, we can obtain

Then, by the maximum principle we infer that $U(x)\geq0$ for all $|x|\geq R$. In addition, by the definition of $U(x)$, obviously, $U(x)\geq0$ for $|x|\leq R$. Thus, we get $U(x)\geq0$ for all $x\in\mathbb{R}^{3}$, furthermore, we have

The proof of Theorem 1.1 is finished.

3.   Conclusions

In this paper, we are interested in the existence and decay property of ground state solutions for a Kirchhoff equation involving fractional Laplacian operator. Since the nonlocal term $\big(\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}} \psi|^{2}\, \mathrm{d} x\big)^{2}$ included in the energy functional $\mathcal{E}(\psi)$ is homogeneous of degree $4$, when $p\leq4$, it brings about two obstacles to the standard mountain-pass arguments both in checking the geometrical assumptions in the corresponding energy functional and in proving the boundedness of the Palais-Smale sequence for $\mathcal{E}(\psi)$. By constructing a Palais-Smale-Pohozaev sequence at the minimax value $c_{mp}$, the existence of ground state solutions to this equation for all $p\in(2, 2_{s}^{*})$ is established by variational arguments. Furthermore, the decay property of the ground state solution is also investigated. Our result extends and improves the recent results in the literature. We believe that the proposed approach in the present paper can also be applied to studying other related variational problems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the reviewers for their valuable comments and suggestions for improvement of the paper.

This work was partially supported by the fund from NSFC (12061212) and the Research Project of Hubei Engineering University (202231).

Conflict of interest

The authors declare that they have no competing interests.