Normalized ground states for fractional Kirchhoff equations with critical or supercritical nonlinearity

1.   Introduction

In this paper, we study mainly the existence of ground states to the Kirchhoff type problem with critical or supercritical nonlinearity

where $\ s\in(\frac{3}{4}, \ 1), \ a, \ b, \ c > 0, \ \frac{6+8s}{3} < \ q < \ 2^{\ast}_{s}, \ p\geq 2^{\ast}_{s}\ (2^{\ast}_{s} = \frac{6}{3-2s}), \ \mu > 0$ is a real parameter, and $(-\bigtriangleup)^{s}$ denotes the fractional Laplacian operator.

The operator $(-\bigtriangleup)^{s}$ can be seen as the infinitesimal generators of Lévy stable diffusion processes, see ^[1,2] for example. This operator appears in several areas such as biology, chemistry and physics (see ^[3,4,5,6]). Problem (1.1) is viewed as being nonlocal because of the appearance of the term $b\int_{\mathbb{R}^{3}}|(-\bigtriangleup)^{\frac{s}{2}}u|^{2}$, which implies that Eq (1.1) is no longer a pointwise identity. This also results in lack of weak sequential continuity of the energy function associated to (1.1), so it make the study of (1.1) particularly interesting. Over the last decade, many mathematicians were particularly keen on the study of nonlinear equations involving nonlocal operators, we can look it up in ^{[7,8,9,10,11,12,13,14]} and the references therein.

It is well known that problem (1.1) arises from looking for the standing wave type solutions $\varphi(x, t) = e^{-i\lambda t}u(x), \ \lambda\in \mathbb{R}$ for the following time-dependent nonlinear fractional Kirchhoff equation

where $0 < s < 1, \ i$ denotes the imaginary unit. The stationary case of (1.2) is the following equation

Clearly, $\varphi$ solves (1.2) if and only if the stand wave $u(x)$ satisfies (1.1) with $f(u) = |u|^{q-2}+\mu|u|^{p-2}$. Alternatively one can consider the existence of normalized solutions to (1.1), that is, solutions with prescribed $L^{2}$-norm. Since solutions $\varphi\in C([0, 1], \ H^s(\mathbb{R}^{3}))$ to (1.2) maintain their mass along time (In fact, multiplying (1.2) by the conjugate $\overline{\varphi}$ of $\varphi,$ integrating over $\mathbb{R}^{3},$ and taking the imaginary part, we get $\frac{d}{dt}|\varphi(t)|_2^2 = 0, \ t\in [0, T].),$ it is natural and interesting, from a physical point view, to search for such solutions.

When $s = 1$, Problem (1.3) becomes the Kirchhoff equation. In the past several years, the Kirchhoff type equations has been studied extensively by many researchers(see ^{[15,16,17,18,19,20,21,22,23]}). For all we know, the existence results to problem (1.1) have been mostly available for the case where $p, q\in(2, 2_{s}^{\ast})$ and $\lambda$ is fixed and assigned. When $a = 1, \ b = 0, \ s = 1$ and $\mu = 0$, i.e., for the Laplacian operator, Jeanjean's ^[24] was the first paper to prove existence of normalized solutions in purely $L^{2}$-supercritical case. Li and Ye in ^[25] considered problem (1.1) with $s = 1, \mu = 0, N = 3, \lambda = -1, q\in(3, 6)$ and proved that (1.1) has at least one least energy solution by dealing with a constrained minimization problem on a manifold of $H^{1}(\mathbb{R}^{3})$, which is obtained by combining the Nehari manifold and the corresponding Pohozaev identity. Liu, Chen and Yang in ^[26] considered problem (1.1) with $2 < q < p < 2_{s}^{\ast}$ and proved some existence results about the normalized solutions. However, there is few literature concerned about the normalized solutions for fractional Kirchhoff equation with critical or supercritical nonlinearity. With regard to the point, we attempt to study this kind of problem in this paper.

It is well known that the fractional order Sobolev space $H: = H^{s}(\mathbb{R}^{3})$ can be defined as follows

endowed with the norm

and the inner product is

According to ^[26], we know that

is also a norm on $H^{s}(\mathbb{R}^{3})$ which is equivalent to $\|u\|_{H}.$ Moreover, we define $H_{r}^{s}(\mathbb{R}^{3}): = \{u\in H^{s}(\mathbb{R}^{3}):u(x) = u(|x|), x\in \mathbb{R}^{3}\}.$

Let $\mathbb{H} = H\times\mathbb{R}$ with the scalar product

and the corresponding norm

Let $\|\cdot\|_{t}$ be the usual norm of space $L^{t}(\mathbb{R}^{3})$ where $2\leq t\leq\infty$. $H$ is continuously embedding into $L^{t}(\mathbb{R}^{3})$ for $t\in [2, 2_{s}^{\ast}]$ and there exists a best constant $S_{s}^{\ast}$ such that

The normalized weak solution for the problem (1.1) is obtained by looking for critical points of the following $C^{1}$ functional

constrained on the $L^{2}$-spheres in $H$:

$u_c$ is called a ground state of (1.1) on $S(c)$ if

Since $p\geq 2_s^*$, the functional $\mathcal{J}_\mu$ is not well defined on $H^{s}(\mathbb{R}^{3})$ unless $p = 2_s^*$. Moreover, we need to overcome the lack of compactness in studying critical and supercritical growth. Hence, we cannot directly use variational methods to prove the existence of normalized solutions. To overcome these difficulties, we use a new method, which came from ^[14,18]. The main idea of this method is to reduce the supercritical problem into a subcritical one. In comparison with previous works, this paper has several new features. Firstly, we consider the nonlinear term with supcritical growth. Secondly, we give the existence of normalized solution for the appropriate truncation problem of (1.1). Finally, the existence of a normalized ground state solution is obtained by Moser iteration method. The results in this paper extend the results in paper ^[4,24,26]. There have been no previous studies considering the existence of normalized ground state solutions for problem (1.1) involving supcritical growth to the best of our knowledge.

Our main result is the following:

Theorem 1.1. For any $c > 0,$ there exists a $\mu^* > 0$ such that, problem (1.1) has a couple of solutions $(u_{c}, \ \lambda_{c})\in H^{s}(\mathbb{R}^{3})\times \mathbb{R}$ for any $\mu \in (0, \mu^*]$. Moreover, $u_{c}$ is a positive ground state, radially symmetric function and $\lambda_{c} < 0.$

Remark 1.2. When $\frac{6+8s}{3} < q < 2_{s}^{\ast}$, $\mathcal{J}_\mu$ is not bounded from below on $S(c)$, i.e., $\inf\limits_{u\in S(c)}\mathcal{J}_\mu(u) = -\infty$. So, the minimization problem constrained on $S(c)$ does not work. We try to look for a critical point with a minimax characterization. Although $\mathcal{J}_\mu$ has a mountain-pass geometry on $S(c)$, the boundedness of the obtained Palais-Smale sequence is not yet clear. Motivated by ^[4], we try to construct an auxiliary map $I_\mu$, which on $S(c)\times\mathbb{R}$ has the same type of geometric structure as $\mathcal{J}_\mu$ on $S(c)$. Besides, the Palais-Smale sequence of $I_\mu$ satisfies the additional condition, which is the key point to obtain the boundedness of the Palais-Smale sequence.

2.   Preliminaries

In this section, we give a truncation argument in order to overcome the lack of compactness in studying critical and supercritical growth. Let $M > 0$ be a constant. For fixed $c > 0, \ \mu > 0, \ M > 0,$ we investigate the existence of ground state for the following truncation problem

where $s\in(\frac{3}{4}, \ 1), \ a, b > 0, \ \frac{6+8s}{3} < \ q < \ 2^{\ast}_{s}, \ p\geq 2^{\ast}_{s}\ (2^{\ast}_{s} = \frac{6}{3-2s}), \ $ and

To investigate (2.1), we define the the energy functional $E_{\mu}:H\rightarrow \mathbb{R}$ by

where $D_{M}(t)\doteq\int_{0}^{t}d_{M}(\tau)d\tau$. It is easy to obtain that $E_{\mu}\in{C^1(H, \mathbb{R})}$ and

for all $u, v \in H$.

Theorem 2.1. For any $c > 0\ $ and $M > 0,$ there exists a $\mu_1 > 0,$ such that, problem (2.1) has a couple of solutions $(u_c, \ \lambda_c)\in H_r^{s}(\mathbb{R}^{3})\times \mathbb{R}$ for any $\mu \in (0, \mu_1]$. Moreover, $u_c$ is a positive ground state, $\lambda_c < 0$ and $E_\mu(u_c) = m_{c, \mu},$ where

and $V(c)$ is the Pohozaev manifold defined in lemma 2.4.

Next, we give some useful preliminary lemmas to prove Theorem 2.1.

Lemma 2.1. ^[8] If $\alpha\in (2, \ 2_{s}^{\ast})$, there exists an optimal constant $C(s, \alpha)$ such that for any $u\in H$,

where $\beta_{\alpha}: = \frac{3(\alpha-2)}{2\alpha}$.

Lemma 2.2. ^[19] $H_r^{s}(\mathbb{R}^{3})$ is compactly embedding into $L^{t}(\mathbb{R}^{3})$ for $t\in (2, 2_{s}^{\ast}).$

As in ^[4], we introduce the useful fiber map preserving the $L^{2}$-norm, that is,

Define the auxiliary functional $I:\mathbb{H}\rightarrow \mathbb{R}$ by

where

then we can obtain that $I_\mu$ is a $C^{1}$-functional.

Lemma 2.3. ^[13] $\text{The map}\ (u, \tau)\in\mathbb{H}\mapsto \tau\star u \in H\ \text{is continuous}.$

Similar to Lemma 2.1 in ^[4], we can easily get the following lemma.

Lemma 2.4. Let $(u, \lambda)\in S(c)\times\mathbb{R}$ be a weak solution of Eq (2.2). Then $u$ belongs to the set

where

Lemma 2.5. For any $u\in S(c)$, $\tau\in\mathbb{R}$ is a critical point for $\Phi_{u}(\tau): = I_\mu(u, \tau)$ if and only if $\tau\star u\in V(c)$.

Proof. For any $u\in S(c)$ and $\tau\in\mathbb{R}$, we have

It is easy to see that Lemma 2.5 holds.

Lemma 2.6. Let $u\in S(c)$ be arbitrary fixed, then

(1) $\int_{\mathbb{R}^{3}}|(-\triangle)^{\frac{s}{2}}(\tau\star u)|^{2}\rightarrow 0$ and $I_\mu(u, \tau)\rightarrow 0$ as $\tau\rightarrow -\infty$;

(2) $\int_{\mathbb{R}^{3}}|(-\triangle)^{\frac{s}{2}}(\tau\star u)|^{2}\rightarrow +\infty$ and $I_\mu(u, \tau)\rightarrow -\infty$ as $\tau\rightarrow +\infty$.

Proof. For fixed $u\in S(c)$, we can easily get the conclusions (1) and (2) from the facts

and $\beta_{r}r\geq \beta_{q}q > 4s$.

Lemma 2.7. For every $u\in S(c)$, there exists a unique $\tau_{u}\in \mathbb{R}$ such that $\tau_{u}\star u\in V(c)$, where $\tau_{u}$ is a strict maximum point for $\Phi_{u}(\tau)$ and $\Phi_{u}(\tau_{u}) > 0$.

Proof. For $u\in S(c)$ and $\tau\in \mathbb{R}$, by (2.9) we have

Since $r\beta_{r}\geq q\beta_{q} > 4s$, it is easy to see that $(\Phi_{u})'(\tau) > 0$ as $\tau\rightarrow -\infty$, and $(\Phi_{u})'(\tau) < 0$ as $\tau\rightarrow \infty$. So, there exists $\tau_{u}\in \mathbb{R}$ such that $(\Phi_{u})'(\tau_{u}) = 0$. From Lemma 2.5, $\tau_{u}\star u\in V(c)$.

Combining with $(\Phi_{u})'(\tau_{u}) = 0$, (2.9) and (2.10), we have

which together with Lemma 2.6 implies that $\tau_{u}$ is unique and it is a strict global maximum point for $\Phi_{u}(\tau)$ and $\Phi_{u}(\tau_{u}) > 0$.

3.   Characterization of mountain pass level

As in ^[4], firstly, we prove that $E_\mu(u)$ has the mountain pass geometry on $S(c)\times\mathbb{R}$ in the following lemma.

Lemma 3.1. There exists $k_{c} > 0$ such that

with

Proof. Let $k > 0$ be arbitrary fixed and suppose $u, v\in S(c)$ are such that

Then for $k$ small enough, by (2.4) and $\frac{3(r-2)}{2s}\geq\frac{3(q-2)}{2s} > 4$, there exist constants $C_1$ and $C_2$ such that

and

By the above inequalities, we can obtain that there exists $k_c > 0$ sufficiently small such that Lemma 3.2 holds.

Next, we need to construct the minimax characterization of $I_\mu$ and $E_{\mu}$.

Lemma 3.2. Let

with

and

with

then we have

Proof. Firstly, we prove that $\tilde{\gamma}_{c, \mu} = \gamma_{c, \mu}.$

For any $\tilde{h}\in \tilde{\Gamma}_{c}$, we can write it into

We set $h(t) = \tilde{h}_{2}(t)\star \tilde{h}_{1}(t)$, then $h(t)\in \Gamma_{c}$, and

which implies $\tilde{\gamma}_{c, \mu}\geq \gamma_{c, \mu}.$ On the other hand, for any $h\in \Gamma_{c}$, if we set $\tilde{h}(t) = (h(t), 0)$, then we get $\tilde{h}\in \tilde{\Gamma}_{c}$ and

This infers that $\tilde{\gamma}_{c, \mu}\leq \gamma_{c, \mu}$. So, $\tilde{\gamma}_{c, \mu} = \gamma_{c, \mu}$.

Secondly, we claim that for $u\in S(c),$ $E_{\mu}(u)\leq 0$ implies $P_{\mu}(u) < 0$.

For $u\in S(c),$ if $E_{\mu}(u)\leq0$, then $\Phi_{u}(0)\leq 0.$ By the proof of Lemma 2.7 and Lemma 2.6, we easily see that $\tau_{u} < 0$, so

That is

Next, we prove that $m_{c, \mu} = \gamma_{c, \mu}$.

For any $u\in V(c)$, by Lemma 2.6 and Lemma 2.3, there exists $t^{-}\ll-1$ and $t^{+}\gg1$ such that

By Lemma 2.7, we have $\max\limits_{\tau\in [0, 1]} E_{\mu}(h_{u}(\tau)) = E_{\mu}(u)$. So we have $m_{c, \mu}\geq\gamma_{c, \mu}$. On the other hand, for any $\tilde{h}(\tau) = (\tilde{h}_{1}(\tau), \tilde{h}_{2}(\tau))\in \tilde{\Gamma}_{c}$, we know that $\tilde{h_{2}}(0)\star\tilde{h_{1}}(0) = \tilde{h_{1}}(0)\in \mathbb{A}_c$, $\tilde{h}_{2}(1)\star\tilde{h}_{1}(1) = \tilde{h}_{1}(1)\in E_\mu^{0}.$ Hence by Lemma 3.1, we can deduce that

and using (3.1),

From Lemma 2.3, the function $\tilde{P}_{\mu}(\tau): = P_{\mu}(\tilde{h}_{2}(\tau)\star\tilde{h}_{1}(\tau))$ is continuous in $[0, 1]$. Therefore, there exists $\bar{\tau}\in (0, 1)$ such that $\tilde{P}_{\mu}(\bar{\tau}) = 0$, which implies that $\tilde{h}_{2}(\bar{\tau})\star\tilde{h}_{1}(\bar{\tau})\in V(c)$, and

which implies that $\gamma_{c, \mu} = \tilde{\gamma}_{c, \mu}\geq m_{c, \mu}$. So, $m_{c, \mu} = \gamma_{c, \mu}$.

Finally, we prove that $m_{c, \mu} > 0$.

For any $u\in V(c)$, then $P_{\mu}(u) = 0$. By (2.4), we have

noticing that $r\beta_{r}\geq q\beta_{q} > 4s$, there exists $\delta > 0$ such that $\inf\limits_{u\in V(c)}\int_{\mathbb{R}^{3}}|(-\triangle)^{\frac{s}{2}}u|^{2}dx\geq \delta$, and

Thus, $m_{c, \mu} > 0$.

Remark 3.3. Let

with

Obviously, $\Gamma_{c}^0\subset \Gamma_{c}, \ E_0(u)\geq E_\mu(u)$ for $u\in S(c).$ Thus we can deduce that $\rho_c$ is independent of positive numbers $\mu, M$ and $\rho_c\geq\gamma_{c, \mu}$ for any $\mu > 0.$

In the following lemma, we give the relationship between the Palais-Smale sequence for $I$ and that of $E_{\mu}$.

Lemma 3.4. Let $\tilde{\gamma}_{c, \mu}$ and $\gamma_{c, \mu}$ be defined in Lemma 3.2. Then there exist a sequence $\{(v_{n}, \tau_{n})\}\subset S(c)\times\mathbb{R}$ such that for $n\rightarrow \infty$, we have

(1) $I_\mu(v_{n}, \tau_{n})\rightarrow \tilde{\gamma}_{c, \mu}$,

(2) $(I_\mu)'|_{S(c)\times\mathbb{R}}(v_{n}, \tau_{n})\rightarrow 0$, i.e., it holds that

and

with

In addition, setting $u_{n}(x) = \tau_{n}\star v_{n}(x)$, then for $n\rightarrow \infty$, we get

(i) $E_{\mu}(u_{n})\rightarrow \gamma_{c, \mu},$

(ii) $P_{\mu}(u_{n})\rightarrow 0,$

(iii) $E'_{\mu}|_{S(c)}(u_{n})\rightarrow 0$, i.e., it holds that

with

Proof. According to the construction of $\tilde{\gamma_{c, \mu}}$, we know that the conclusions (1) and (2) follow directly from the Ekeland's Variational Principle [8, Proposition 2.2]. Next we mainly show (i)–(iii).

For (i), by Lemma 3.2, $\tilde{\gamma}_{c, \mu} = \gamma_{c, \mu}$. we notice that

thus (i) holds.

By (2.9), we can get that $\partial_{\tau}I_\mu(v_{n}, \tau_{n}) = sP_{\mu}(\tau_{n}\star v_{n}).$ Thus, (ii) is a consequence of $\partial_{\tau}I_\mu(v_{n}, \tau_{n})\rightarrow 0$ as $n\rightarrow \infty$.

For the proof of (iii), by the definition of $I_\mu$, we have

where $\tilde{\varphi}\in T_{v_{n}}.$

On the other hand, for any $\varphi$ with satisfying $\varphi\in T_{u_{n}},$ by using (2.3), we have

Setting

we get (iii) if we could show that $\tilde{\varphi}\in T_{v_{n}}$. In fact, $\tilde{\varphi}\in T_{v_{n}}$ follows from the following equalities

4.   Proof of Theorem 2.1

According to Lemma 3.4 and Lemma 3.2, there exist a Palais-Smale sequence $\{u_{n}\}\subset S(c)$ for $E_{\mu}|_{S(c)}$ at level $\gamma_{c, \mu} > 0$, and it satisfies $P_{\mu}(u_{n})\rightarrow 0$ as $n\rightarrow \infty$. By applying the Lagrange multipliers rule there exists $\{\lambda_{n}\}\subset \mathbb{R}$ such that

(1). As $P_{\mu}(u_{n})\rightarrow 0$, we have

Thus, by (4.2) we deduce that

Since $\frac{6+8s}{3} < q < 2_{s}^{\ast}$ and (2.7), it implies that $\frac{s}{\beta_{q}q} < \frac{1}{4}$ and $\beta_{r}r\geq\beta_{q}q$. According to (4.3), we can deduce the boundedness of $\int_{\mathbb{R}^{3}}|(-\triangle)^{\frac{s}{2}}u_{n}|^{2}$, thus $\{u_{n}\}$ is bounded in $H$.

(2). According to Lemma 2.2, we know that the embedding $H_{r}^{s}(\mathbb{R}^{3})\hookrightarrow L^{t}(\mathbb{R}^{3})$ is compact for $t\in (2, \ 2_{s}^{\ast})$, and we can deduce that there exists $u_c\in H_{r}^{s}(\mathbb{R}^{3})$ such that, up to a subsequence, $u_{n}\rightharpoonup u_c$ weakly in $H$, $u_{n}\rightarrow u_c$ strongly in $L^{q}(\mathbb{R}^{3})$ for $q\in (\frac{6+8s}{3}, 2_{s}^{\ast})$. since $\{u_{n}\}\subset S(c)$ is bounded in $H$. By (4.1), we obtain that

Using the fact that the boundedness of $\{u_{n}\}$ in $H$ and (4.2), we can deduce that $\{\lambda_{n}\}$ is bounded. Hence, up to a subsequence $\lambda_{n}\rightarrow \lambda_c\in \mathbb{R}$.

(3). We claim that $u_c \not\equiv 0$. We assume by contradiction that $u_c\equiv0$, by (4.2) we deduce that $\int_{\mathbb{R}^{3}}|(-\triangle)^{\frac{s}{2}}u_{n}|^{2} \rightarrow 0$. Recalling that $P_{\mu}(u_{n})\rightarrow 0$, according to (4.3), we have $E_{\mu}(u_{n})\rightarrow 0$, which is a contradiction to the assumption that $E_{\mu}(u_{n})\rightarrow \gamma_{c, \mu}\neq 0$. Now, since $\lambda_{n}\rightarrow \lambda_c$ and $u_{n}\rightarrow u_c\neq 0$ weakly in $H$, together with (4.1), we know $(u_c, \ \lambda_c)$ is a couple of solutions to (2.1). By the Pohozaev identity, we obtain

Combining with the (4.4) for $u_c$, we get

Since $\frac{6+8s}{3} < q < 2_{s}^{\ast}$ and (4.5), there exists $\mu_1 > 0$ such that $\lambda_c < 0$ for $\mu\in (0, \mu_1]$.

(4). Testing (4.1) and (2.1) with $u_{n} -u_c$, we can obtain that

Using the strong $L^{p}$ convergence of $u_{n}$, we infer that

which, being $\lambda_c < 0$, implies $u_{n}\rightarrow u_c$ strongly in $H$. Therefore, $E_{\mu}(u_{n})\rightarrow E_{\mu}(u_c), \ \text{as}\ n\rightarrow \infty.$ From Lemma 2.4 and Lemma 3.2, we easily obtain that $u_c$ is a ground state of (2.1) and $E_{\mu}(u_c) = m_{c, \mu}.$

5.   Proof of main result

In this section, we devote to complete the proof of Theorem 1.1. From the truncation argument in Sections 2–4, we can see that if the ground state $u_c$ of (2.1) satisfy $\|u_c\|_\infty \leq M.$ Then $u_c\in H$ is a ground state of (1.1).

Lemma 5.1. Let$(u_{c}, \ \lambda_{c})$ be a couple of solutions of problem (2.1) for $\mu\in (0, \mu_1]$, then there exists a constant $K_c > 0$ independent of $\mu, M > 0$ such that $\|u_c\|\leq K_c$.

Proof. By Theorem 2.1 and Lemma 2.4, it is easy to see that

It follows from (5.1) and Remark 3.3 that

Consequently, there exists a constant $K_c > 0$ independent of $\mu, M > 0$ such that $\|u_c\|\leq K_c$.

Lemma 5.2. If $(u_{c}, \ \lambda_{c})$ be a couple of solutions of problem (2.1) for $\mu\in (0, \mu_1]$, then $u_c\in L^{\infty}(\mathbb{R}^{3})$, and there exists a constant $B_c > 0$ independent $\mu, M > 0$ such that

Proof. For convenience, we replace $u_c$ with $u$ in the following. Let $L > 0$ and $\beta > 1$, we first define the following functions:

where $u_{L} = min\{u, L\}$. Since $\Upsilon$ is an increasing function, we have

Let $\Phi(t) = \frac{1}{2}|t|^{2}$ and $\Psi(t) = \int_{0}^{t}(\Upsilon'(\tau))^{\frac{1}{2}}d\tau$.Then, if $x > y$, by Cauchy-Schwarz inequality, we have

The same arguments hold for $x\leq y$. Therefore,

By the definition of $u_{L}$, it is easy to see that $|uu_{L}^{2(\beta-1)}|\leq L^{2(\beta-1)}u$ and $\Upsilon(u)\in H$. Taking $\Upsilon(u)$ as a test function in Eq (2.1), and let $g_{\mu, M}(x, t) = |t|^{q-2}t+\mu d_{M}(t)$, we obtain

Since $\Psi(u)\geq \frac{1}{\beta}uu_{L}^{\beta-1}$ and (5.2), we get

For any $\varepsilon > 0$, there exists $C_{\varepsilon} > 0$ such that

Let $\omega_{L} = uu_{L}^{\beta-1}$. By employing Hölder's inequality and (5.3)–(5.5), we have

where $\frac{q-2}{2_{s}^{\ast}}+\frac{1}{t} = 1$ and $2t\in (2, 2_{s}^{\ast})$. Moreover, it follows from (1.4) that

Therefore, we deduce from (5.6) and (5.7) that

where $C > 0$ is a constant. From the definition of $u_{L}$, we have $u_{L}\leq u$ in $\mathbb{R}^{3}$. Letting $L\rightarrow +\infty$, using the Fatou's Lemma, one has

By the interpolation inequality, we get $\|u\|_{2\beta}\leq \|u\|_{2}^{1-\sigma}\|u\|_{2\beta t}^{\sigma}$, where $\sigma\in (0, 1)$ satisfies $\frac{1}{2\beta} = \frac{1-\sigma}{2}+\frac{\sigma}{2\beta t}$. Thus, $\sigma = \frac{t(\beta-1)}{t\beta-1}$, which shows that $\sigma\rightarrow1$ as $\beta\rightarrow +\infty$. Since $2\beta(1-\sigma) = 2+\frac{2(1-\beta)}{t\beta-1} < 2$, we get

which together with (5.8) yields

where $k\in \{\sigma, 1\}$ and $C_{1} > 0$ is a constant. Let $\theta: = \frac{2_{s}^{\ast}}{2t}$, then $\theta > 1$. Taking $\beta = \theta$ in (3.13), we deduce that

where $k_{1}\in \{\sigma_{1}, 1\}$ and $\sigma_{1} = \frac{t(\theta-1)}{t\theta-1}$. Taking $\beta = \theta^{2}$ in (5.9), we get

where $k_{2}\in \{\sigma_{2}, 1\}$ and $\sigma_{2} = \frac{t(\theta^{2}-1)}{t\theta^{2}-1}$. Combining (5.10) with (5.11), we have

Taking $\beta = \theta^{i}, i\in \mathbb{N}$, one has

where $k_{i}\in \{\sigma_{i}, 1\}$ and $\sigma_{i} = \frac{t(\theta^{i}-1)}{t\theta^{i}-1}$.

Next, we divide into two cases: $\|u\|_{2_{s}^{\ast}}\geq1$ and $\|u\|_{2_{s}^{\ast}} < 1$.

(1) Assume that $\|u\|_{2_{s}^{\ast}}\geq1$ is in force. In view of $k_{1}k_{2}\ldots k_{i}\leq1$, we have $\|u\|_{2_{s}^{\ast}}^{k_{1}k_{2}\ldots k_{i}}\leq \|u\|_{2_{s}^{\ast}}$. Letting $i\rightarrow +\infty$ in (5.12), we can know that

(2) Assume that $\|u\|_{2_{s}^{\ast}} < 1$ is true. By $\sigma_{i} = \frac{t(\theta^{i}-1)}{t\theta^{i}-1} = 1-\frac{t-1}{t\theta^{i}-1}$ and $k_{i}\in {\sigma_{i}, 1}$, we have $0 < \sigma_{1}\sigma_{2}\ldots\sigma_{i}\leq k_{1}k_{2}\ldots k_{i}\leq1$, which shows that $\sum\limits_{m = 1}^{i}ln \sigma_{m}\leq\sum\limits_{m = 1}^{i}ln k_{m}\leq0$. From the fact that $ln(1-s)\geq\frac{-s}{1-s}$ for all $s\in (0, 1)$, one has

which implies that

By $\|u\|_{2_{s}^{\ast}} < 1$, we have $\|u\|_{2_{s}^{\ast}}^{k_{1}k_{2}\ldots k_{i}}\leq \|u\|_{2_{s}^{\ast}}^{ e^{A}}$. Similarly, letting $i\rightarrow +\infty$ in (3.16), we reach

Consequently, we have $u\in L^{\infty}(\mathbb{R}^{3})$ and

where $\tau = 1$ or $\tau = e^{A}\leq1$.

Finally, by (1.4) and Lemma 5.1, there exists $C_{2} > 0$ such that $\|u\|_{2_{s}^{\ast}}\leq C_{2}$. Therefore, it follows from (5.13) and $\theta = \frac{2_{s}^{\ast}-q+2}{2}$, there exists a constant $B_c > 0$ independent $\mu, \ M > 0$ such that

Proof of the Theorem 1.1. By Lemma 5.2, for any $c > 0,$ there exists a constant $B_c > 0$ independent on $\mu$ and $M$ such that

Thus, for large $M > 0,$ we can choose small $\mu^* > 0$ with $\mu^*\leq \mu_1$ such that $\|u_c\|_\infty \leq M$ for all $\mu\in(0, \mu^*].$ By Theorem 2.1, problem (1.1) has a couple of solutions $(u_{c}, \ \lambda_{c})\in H_r^{s}(\mathbb{R}^{3})\times \mathbb{R}$ for any $\mu \in (0, \mu^*]$. Moreover, $u_{c}$ is a positive ground state, radially symmetric function and $\lambda_{c} < 0.$

Acknowledgments

The authors would like to thank the anonymous referees for carefully reading this paper and making valuable comments and suggestions. This research was funded by the National Natural Science Foundation of China (61803236) and Natural Science Foundation of Shandong Province (ZR2018MA022).

Conflict of interest.

All authors declare no conflicts of interest in this paper.