Multiple positive solutions for a bi-nonlocal Kirchhoff-Schr$\ddot{\mathrm{o}}$dinger-Poisson system with critical growth

1.   Introduction and main result

This paper is concerned with the following Kirchhoff-Schr$\ddot{\mathrm{o}}$dinger-Poisson system:

where $\Omega\subset \mathbb{R}^3$ is a smooth bounded domain, $\lambda > 0$, $0\leq r < 1$, $0 < s < \frac{1-r}{3(r+1)}$ and $F(x, u) = \int_{0}^uf(x, \xi)d\xi$. We assume that $f\in C^1(\overline{\Omega}\times\mathbb{R}, \mathbb{R})$ and there exist constants $a_1, a_2 > 0$ and $\frac{6(r+1)}{r+2} < q < \frac{4}{s+1}, 0 < s < \frac{1-r}{3(r+1)}$, such that

When $\left(\int_{\Omega}|\nabla u|^2dx\right)^r = 1$ and $s = 0$, the system (1.1) reduces to the boundary value problem

Problem (1.3) has been extensively studied, by using variational methods and critical point theory under suitable assumptions on $V$, $f$; see ^{[1,2,3,4,5,6,7]} and the references therein.

On the other hand, considering just the first equation in (1.1) with the potential equal to zero, we have the problem

where $a, b > 0$, which was proposed by Kirchhoff in ^[8] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. The appearance of the nonlocal term $\int_{\Omega}|\nabla u|^2dx$ in the equations makes them important in many physical applications. We have to point out that such nonlocal problems appear in other fields like biological systems, such as population density, where $u$ describes a process which depends on the average of itself (see ^[9]). The Kirchhoff type problem (1.4) with critical growth began to call the attention of researchers; we can see ^{[10,11,12,13,14,15,16,17,18]} and the references therein.

In particular, Che et al. in ^[19] considered the following Kirchhoff-Schr$\ddot{\mathrm{o}}$dinger-Poisson system with critical growth:

where $a > 0, b\geq0, q\in[4, 6)$, and $\lambda > 0$ is a parameter. Under some suitable conditions on $V(x)$, $g(x)$ and $h(x)$, by using the Nehari manifold technique and Ljusternik-Schnirelmann category theory, they established the number of positive solutions with the topology of the global maximum set of $h$ when $\lambda$ is small enough. Furthermore, with the aid of the mountain pass theorem, they obtained an existence result for $\lambda$ sufficiently large.

In ^[20], Chabrowski investigated the bi-nonlocal problem for the nonlinear elliptic equation of the form

where $2 < p\leq2^{*}, 2 < q < 2^{*}, s > 0, r > 0$, and $2^* = \frac{2N}{N-2} (N\geq3)$ denotes the critical Sobolev exponent. The existence of solutions in critical and subcritical cases is obtained by using the variational method. A similar problem with Dirichlet boundary conditions has been considered in ^[21]. Motivated by the above references, we study the existence of multiple positive solutions for system (1.1). Our main difficulties are as follows: The critical growth of system (1.1) leads to the lack of compactness of the embedding $H_0^1(\Omega)\hookrightarrow L^6(\Omega)$, and it is difficult to prove the energy functional belongs to the range where the (PS) condition holds. We overcome this difficulty by using the concentration compactness principle.

Throughout this paper, we make use of the following notations:

● The space $H_0^1(\Omega)$ is equipped with the norm $\|u\| = (\int_{\Omega}|\nabla u|^2dx)^{\frac{1}{2}}$, and the norm in $L^p(\Omega)$ is denoted by $|u|_{p} = (\int_{\Omega}|u|^pdx)^{\frac{1}{p}}$;

● $C, C_1, C_2, \dots$ denote various positive constants, which may vary from line to line;

● We denote by $S_\rho$ (respectively, $B_\rho$) the sphere (respectively, the closed ball) of center zero and radius $\rho$, i.e., $S_\rho = \{u\in H_0^{1}(\Omega): \|u\| = \rho\},$ $B_\rho = \{u\in H_0^{1}(\Omega): \|u\|\leq \rho\}$;

● $\rightarrow$ (respectively, $\rightharpoonup$) denotes strong (respectively, weak) convergence;

● Let $S$ be the best Sobolev constant, namely,

Our main result is the following:

Theorem 1.1. Assume that $q(s+1) < 4\leq2(r+1) < 6$, $\lambda > 0$, and $0 < s < \frac{1-r}{3(r+1)}$. Then, there exists $\Lambda_{*} > 0$ such that for any $\lambda\in(0, \Lambda_{*})$, system (1.1) has at least two positive solutions.

Remark 1.1. As we shall see, in the system (1.1), when $f(x, u) = |u|^{q-2}u$, $F(x, u) = |u|^{q}$, Chabrowski established the existence of solutions. In this paper, due to the nonlocal term $\phi u$ in (1.1), new treatments are needed for our problem. Therefore, in this article, we extend the relevant results of ^[20].

2.   Proof of Theorem 1.1

First, by using the Lax-Milgram theorem, for each $u\in H_0^{1}(\Omega)$, there exists a unique solution $\phi_u \in H_0^{1}(\Omega)$ which satisfies the second equation of system (1.1). We substitute $\phi_u$ into the first equation of system (1.1), and then system (1.1) is transformed into the following problem:

We define the energy functional corresponding to problem (2.1) by

We say that a function $u\in H_0^1(\Omega)$ is called a weak solution of problem (2.1) if for every $\varphi\in H_0^1(\Omega)$, there holds

Before proving Theorem 1.1, we give the following important Lemma.

Lemma 2.1. (See ^[22,23]) For all $u\in H_0^1(\Omega)$, there exists a unique solution $\phi_u \in H_0^{1}(\Omega)$ of

and

$(1)$ $\|\phi_u\|^2 = \int_{\Omega}\phi_uu^2dx$;

$(2)$ $\phi_u\geq0$. Moreover, $\phi_u > 0$ when $u\neq0$;

$(3)$ For each $t\neq0$, $\phi_{tu} = t^2\phi_{u}$;

$(4)$

$(5)$ If $u_n\rightharpoonup u$ in $H_0^1(\Omega)$, then $\phi_{u_n}\rightarrow \phi_u$ in $H_0^1(\Omega)$, and

Lemma 2.2. There exist constants $\delta, \rho, \Lambda_0 > 0$, for all $\lambda\in (0, \Lambda_0)$ such that the functional $I_{\lambda}$ satisfies the following conditions:

$(\mathrm{i})$ $I_{\lambda}|_{u\in S_\rho}\geq \delta > 0$, $\inf_{_{u\in B_\rho}}I_{\lambda}(u) < 0$;

$(\mathrm{ii})$ There exists $e\in H_0^{1}(\Omega)$ with $\|e\| > \rho$ such that $I_{\lambda}(e) < 0$.

Proof. $(\mathrm{i})$ According to H$\ddot{\mathrm{o}}$lder's inequality and (1.5), one has

By (1.2), we have

and

Therefore, it follows from (1.5), (2.2) and (2.4) that

Let $H(t) = \frac{1}{2(r+1)}t^{2(r+1)-q(s+1)}-\frac{1}{6}S^{-3}t^{6-q(s+1)}$ for $t > 0$, and then there exists

such that $\max_{t > 0}H(t) = H(\rho) > 0$. Setting

there exists a constant $\delta > 0$, such that $I_{\lambda}|_{u\in S_{\rho}}\geq \delta$ for each $\lambda\in (0, \Lambda_0)$. Moreover, for every $u\in H_0^{1}(\Omega)\backslash\{0\}$, we get

So, we obtain $I_{\lambda}(\tau u) < 0$ for all $u\neq 0$ and $\tau$ small enough. Hence, for $\|u\|$ small enough, we have

$(\mathrm{ii})$ Set $u\in H_0^{1}(\Omega)$, and we get

as $\tau\rightarrow +\infty$, which implies that $I_{\lambda}(\tau u) < 0$ for $\tau > 0$ large enough. Consequently, we can find $e\in H_0^{1}(\Omega)$ with $\|e\| > \rho$ such that $I_{\lambda}(e) < 0$. The proof is complete.

Lemma 2.3. Assume that $\lambda > 0$, $q(s+1) < 4\leq2(r+1) < 6$, and $0 < s < \frac{1-r}{3(r+1)}$. Then, the functional $I_{\lambda}$ satisfies the $(PS)_{c_{\lambda}}$ condition for each

where

Proof. Let $\{u_n\}\subset H_0^{1}(\Omega)$ be a $(PS)$ sequence for $I_{\lambda}$ at the level $c_{\lambda}$, that is,

Combining with (2.2)–(2.4), we get

Then, this implies that $\{u_n\}$ is bounded in $H_0^{1}(\Omega)$ for all $q(s+1) < 4\leq2(r+1) < 6$. Thus, we may assume up to a subsequence, still denoted by $\{u_n\}$, there exists $u\in H_0^{1}(\Omega)$ such that

as $n\rightarrow \infty$. Next, we prove that $u_n\rightarrow u$ strongly in $H_0^{1}(\Omega)$. By using the concentration compactness principle (see ^[24]), there exist an at most countable set $J$, a family of points $\{x_j\}_{j\in J}\subset \bar{\Omega}$, and positive numbers $\{\nu_j\}_{j\in J}$, $\{\mu_j\}_{j\in J}$ such that

Moreover, we have

Let $\varphi_{\varepsilon, j}(x)$ be a smooth cut-off function centered at $x_j$ such that $0\leq\varphi_{\varepsilon, j}\leq 1,$ $|\nabla \varphi_{\varepsilon, j}|\leq\frac{2}{\varepsilon}$, $\varepsilon > 0$, and

Noting that $\{u_n\varphi_{\varepsilon, j}\}$ is bounded in $H_0^1(\Omega)$ uniformly for $n$, combining with (2.6) and (2.8), we have

Similarly, we can obtain

By using the H$\ddot{\mathrm{o}}$lder inequality and $|\nabla \varphi_{\varepsilon, j}|\leq\frac{2}{\varepsilon}$, there exists $C_2 > 0$, and we have

We also derive that

and

By (2.5) and (2.9)–(2.13), we get

that is, $\nu_{j}\geq \mu_{j}^{r+1}$. If $\nu_{j} > 0$, by (2.7), we obtain

Now, we show that (2.14) is impossible. Assume that there exists $j_0\in J$, such that $\mu_{j_0}\geq S^\frac{3}{2-r}$ and $x_{j_0}\in \Omega$. It follows from (2.2)–(2.5) that

Let

It is clear that $\lim_{t\rightarrow0}G(t) = 0$, and $\lim_{t\rightarrow +\infty}G(t) = +\infty$. Therefore, there exists $T > 0$ such that $G(T) = \min_{t\geq0}G(t)$, that is,

From (2.15) we obtain

and by simple calculation, we have

Hence, we can see that

where

We obtain that $c_\lambda\geq c_*$. This is a contradiction, which indicates that $\nu_j = \mu_j = 0$ for every $j\in J$, which implies that $u_n\rightarrow u$ in $L^6(\Omega)$. We may assume that

Combining with (2.5) and (2.6), we have

Hence, we obtain

which implies $u_n\rightarrow u$ in $H_0^1(\Omega)$. The proof is complete.

Choose the extremal function

satisfying

Let $\Psi\in C^1(\mathbb{R}^3)$ such that $\Psi(x) = 1$ on $B_{\frac{R}{2}}(0)$, $\Psi(x) = 0$ on $\mathbb{R}^3\setminus B_R(0)$, and $0\leq\Psi(x)\leq 1$ on $\mathbb{R}^3$. Set $u_\varepsilon(x) = \Psi(x)U_{\varepsilon}(x)$. From ^[25], one has

and

Then, we have the following Lemma.

Lemma 2.4. Suppose that $\lambda > 0$, $q(s+1) < 4\leq2(r+1) < 6$, and $0 < s < \frac{1-r}{3(r+1)}$. Then,

Proof. According to the definition of $u_\varepsilon$ and (2.4), it holds that

From Lemma 2.1 and (2.16), we have the following estimate:

Since $I_{\lambda}(tu_{\varepsilon})\rightarrow -\infty$ as $t\rightarrow \infty$, by Lemma 2.2, there exists $t_{\varepsilon} > 0$ such that

Moreover, by the continuity of $I_{\lambda}$, there exist positive constants $t_1$ and $t_2$ such that $0 < t_1\leq t_{\varepsilon}\leq t_2 < +\infty$. As a consequence of the above fact, one has

We have used the fact that $\frac{6(r+1)}{r+2} < q < \frac{4}{s+1}$ and let $\varepsilon = \lambda^{\frac{2(r+1)}{2(r+1)-q(s+1)}}$.

and then

The proof is complete.

Lemma 2.5. Suppose that $0 < \lambda < \Lambda_0$ ($\Lambda_0$ is as in Lemma 2.2). Then, system (1.1) has a positive solution $u_\lambda$ satisfying $I_\lambda(u_\lambda) < 0$.

Proof. It follows from Lemma 2.2 that

By the Ekeland variational principle ^[26], there exists a minimizing sequence $\{u_n\}\subset {\overline{B_\rho(0)}}$ such that

Therefore, we obtain that $I_\lambda(u_n)\rightarrow m$ and $I_\lambda'(u_n)\rightarrow 0$ as $n\rightarrow \infty$. Since $\{u_n\}$ is a bounded sequence, and ${\overline{B_\rho(0)}}$ is a closed convex set, we may assume up to a subsequence, still denoted by $\{u_n\}$, there exists $u_{\lambda}\in{\overline{B_\rho(0)}}\subset H_0^{1}(\Omega)$ such that

By the lower semi-continuity of the norm with respect to weak convergence, we get

Thus, $I_{\lambda}(u_\lambda) = m < 0$, and we can see that $u_\lambda\not\equiv0$. $I_\lambda(|u_\lambda|) = I_\lambda(u_\lambda)$, which suggests that $u_\lambda\geq0$. Therefore, by the strong maximum principle, we obtain that $u_\lambda$ is a positive solution of system (1.1). The proof is complete.

Lemma 2.6. Assume that $0 < \lambda < \Lambda_{*}$ $(\Lambda_{*} = \min\{\Lambda_0, \Lambda_1\})$. Then, the system (1.1) has a positive solution $u_{\ast}\in H_0^1(\Omega)$ with $I_{\lambda}(u_{\ast}) > 0$.

Proof. From the mountain pass lemma and Lemma 2.2, there exists a sequence $\{u_n\}\subset H_0^1(\Omega)$ such that

where

and

According to Lemma 2.3, we know that $\{u_n\}\subset H_0^1(\Omega)$ has a convergent subsequence, still denoted by $\{u_n\}$, such that $u_n\rightarrow u_{*}$ in $H_0^1(\Omega)$ as $n\rightarrow \infty$.

which implies that $u_{*}\not\equiv0$. It is similar to Lemma 2.5 that $u_\ast > 0$, that is, $u_{\ast}$ is a positive solution of system (1.1) such that $I_{\lambda}(u_{*}) > 0$. The proof is complete.

3.   Conclusions

In this paper, we considered a class of bi-nonlocal Kirchhoff-Schr$\ddot{\mathrm{o}}$dinger-Poisson system with critical growth. Under some suitable assumptions, by using the concentration compactness principle, we obtained the multiplicity of positive solutions.

Acknowledgments

The authors thank the anonymous referees for careful reading and some helpful comments, which greatly improved the manuscript. This work is supported by the National Natural Science Foundation of China (No.11661021; No.11861021).

Conflict of interest

The authors declare there are no conflicts of interest.