Multiple solutions for a fractional p-Kirchhoff equation with critical growth and low order perturbations

1.   Introduction and main result

Consider the following fractional $p$-Kirchhoff type equation with critical growth

where $\Omega\subset \mathbb{R}^N$ is a smooth bounded domain containing $0$, $(-\Delta)_p^s$ denotes the fractional $p$-Laplacian, $M(t)=a+bt^{k-1}$ for $t\geq0$ and $k>1$, $a, b>0$, $\lambda>0$ is a parameter, $0<s<1$, $0\leq\alpha<ps<N$, $\frac{N(p-2)+ps}{p-1}<\beta<\frac{N(p_\alpha^*-1)+\alpha}{p_\alpha^*}$, $1<p<pk<p_\alpha^*=\frac{p(N-\alpha)}{N-ps}$ is the fractional critical Hardy-Sobolev exponent.

Problem (1.1) reduces to the following stationary analogue of the Kirchhoff equation

which was proposed by Kirchhoff in ^[12] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings

Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. Here, $L$ is the length of the string, $f(x,u)$ is the area of the cross section, $E$ is the Young modulus of the material, $\rho$ is the mass density and $P_0$ is the initial tension. The appearance of nonlocal term $\int_{\Omega}|\nabla u|^2dx$ in the equations make its importance in many physical applications. It was pointed 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 ^[1]).

Recently a great attention has been focused on studying the problems involving fractional Sobolev spaces and corresponding nonlocal equations. Indeed, nonlocal fractional problems arise in a quite natural way in many different contexts, such as, optimization, finance, phase transitions, stratified materials, anomalous diffusion, semipermeable membranes and flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, water waves and so on, we refer to ^[15] for more details.

In particular, Chen et al. in ^[6] considered the following fractional $p$-Laplacian equation with subcritical and critical growths

where $0<s<1, p>1, \lambda, \mu>0, 0\leq\alpha\leq ps<N, p\leq r\leq p_\alpha^*$. They obtained the existence of positive solutions, ground state solutions and sign-changing solutions of the fractional $p$-Laplacian Eq (1.4) by using the variational method.

In ^[22], Xiang et al. studied the following fractional $p$-Laplacian Kirchhoff type equation with critical growth

where $M(t)=a+bt^{k-1}$ for $t\geq0$ and $k>1$, $a\geq0$, $b>0$, $0<s<1$ and $1<p<N/s$, $p_s^*=Np/(N-ps)$ is the critical Sobolev exponent, $\lambda\geq0$ is a parameter, $f\in L^{p_s^*/(p_s^*-1)}(\mathbb{R}^N\backslash\{0\})$ is a nonnegative function. By the variational method, the authors proved that Eq (1.5) admits at least two nonnegative solutions.

Recently, the following fractional $p$-Laplacian Kirchhoff type equation with critical growth has been well studied by various authors

where $M(t)=a+bt^{k-1}$ with $t\geq 0$ and $k>1$, $0<s<1$, $a\geq0, b>0$, $0\leq \alpha<ps<N$, $\lambda>0$ is a parameter. When $f(x,u)=w(x)|u|^{q-2}u$, Fiscella and Pucci in ^[10] deal with the existence and the asymptotic behavior of nontrivial solutions for Eq (1.6) with $pk<q<p_\alpha^*$. In ^[5], Chen and Gui obtained the existence of multiple solutions to Eq (1.6) with $w(x)=1$ and $1<q<p<pk$. When $f(x,u)=(I_\mu\ast F(x,u))f(u)$, $I_\mu=|x|^{-\mu}$ is the Riesz potential of order $\mu\in (0, \min\{N, 2ps\})$, Chen ^[4] established the existence of positive solutions to Eq (1.6). Chen, Rǎdulescu and Zhang in ^[7] obtained the existence of a positive weak solution of Eq (1.6) with $f(x,u)=|u|^{q-2}u$ and $1<q<\frac{Np}{N-ps}$. Many papers studied the existence of infinitely many weak solutions and nontrivial solutions of Eq (1.6), we refer the readers to ^{[3,9,11,16,17,18,19,20,21,23,24]} and the reference therein.

In this paper, we are interested in the existence and multiplicity of positive solutions of Problem (1.1) with critical growth. Our technique based on the Ekland variational principle and the Mountain pass lemma. Since the Problem (1.1) is critical growth, which leads to the cause of the lack of compactness of the embedding $W^{s,p}(\Omega)\hookrightarrow L^{p_\alpha^*}(\Omega)$, we overcome this difficulty by using the concentration compactness principle.

Now we state our main result.

Theorem 1.1. Let $0\leq\alpha<ps<N$ and $\frac{N(p-2)+ps}{p-1}<\beta<\frac{N(p_\alpha^*-1)+\alpha}{p_\alpha^*}$, there exists $\Lambda_{\ast}>0$ such that for all $\lambda\in(0,\Lambda_{\ast})$, problem (1.1) has at least two positive solutions.

2.   Preliminary results

Define $W^{s,p}(\Omega)$, the usual fractional Sobolev space endowed with the norm

Let $Q=\mathbb{R}^N\times\mathbb{R}^N\setminus (\mathcal{C}\Omega\times\mathcal{C}\Omega)$ with $\mathcal{C}\Omega=\mathbb{R}^N\setminus \Omega$, define

The space $X$ is endowed with the norm

where the norm in $L^p(\Omega)$ is denoted by $\|\cdot\|_p$. The space $X_0$ is defined as $X_0=\{u\in X: u=0\; {\rm{on}}\; {\mathcal{C}\Omega}\}$ or equivalently the closure of $C_0^{\infty}$ in $X$, for all $p>1$, it is a uniformly convex Banach space endowed with the norm

The dual space of $X_0$ will be denoted by $X^{*}_0$. Since $u=0$ in $\mathbb{R}^N\setminus \Omega$, the integral in (2.1) can be extended to $\mathbb{R}^N\times\mathbb{R}^N$.

The energy functional $I_\lambda: X_0 \rightarrow \mathbb{R}$ associated with Eq (1.1) is

We say that $u$ is a weak solution of Eq (1.1), if $u$ satisfies

for all $\varphi\in X_0$, where

Let $S_\alpha$ be the best fractional critical Hardy-Sobolev constant

Denote by $S_\rho$ (respectively, $B_\rho$) the sphere (respectively, the closed ball) of center zero and radius $\rho$, i.e., $S_\rho=\{u\in X_0: \|u\|=\rho\},$ $B_\rho=\{u\in X_0: \|u\|\leq \rho\}$. $C, C_1, C_2,\dots$ denote various positive constants, which may vary from line to line.

Then, we can obtain the following useful Lemma.

Lemma 2.1. There exist constants $r,\rho,\Lambda_0>0$, such that the functional $I_\lambda$ satisfies the following conditions for all $\lambda\in(0,\Lambda_0):$

$(\mathrm{i})$ $I_\lambda(u)\geq r$ with $\|u\|=\rho$ and $\inf I_\lambda(u)<0$ for $\|u\|\leq \rho;$

$(\mathrm{ii})$ There exists $e\in X_0$ such that $\|e\|>\rho$ and $I_\lambda(e)<0$.

Proof. $(\mathrm{i})$ Let $R$ be a constant such that $\Omega\subset B(0,R)=\{x\in \mathbb{R}^N: |x|<R\}$, by the H$\ddot{\mathrm{o}}$lder inequality and the Sobolev inequality, for all $\beta<\frac{N(p_\alpha^*-1)+\alpha}{p_\alpha^*}$, we have

where $\psi=\frac{\omega}{\alpha\left(\frac{ p_\alpha^*}{p_\alpha^*-1}-1\right)+N-\frac{\beta p_\alpha^*}{p_\alpha^*-1}}R^{\alpha\left(\frac{ p_\alpha^*}{p_\alpha^*-1}-1\right)+N-\frac{\beta p_\alpha^*}{p_\alpha^*-1}}$ and $\omega$ denotes the $N$-dimensional measure of the unit sphere. Combining with (2.2) and the Sobolev inequality, one has

Let $h(t)=\frac{b}{pk}t^{pk-1}-\frac{1}{p_\alpha^*}S_\alpha^{-{\frac{p_\alpha^*}{p}}}t^{{p_\alpha^*}-1}$ for $t>0$, then there exists

such that $\max_{t>0}h(t)=h(\rho)>0$. Setting $\Lambda_0=\frac{S_\alpha^{\frac{1}{p}}h(\rho)}{\psi^\frac{p_\alpha^*-1}{p_\alpha^*}}$, there exists a constant $r>0$, for all $\lambda \in (0,\Lambda_0)$, we obtain that $I_\lambda(u)\geq r>0$ with $\|u\|=\rho$.

For all $u \in X_0\backslash${0}, we get

Hence, we obtain that $I_\lambda(tu)<0$ with $t>0$ small enough, when $\|u\|\leq\rho$, one has

$(\mathrm{ii})$ For every $u \in X_0\backslash${0}, we have

as $t\rightarrow +\infty$. Consequently, we can find $e \in X_0$ such that $I_\lambda(e)<0$ provided with $\|e\|>\rho$. The proof is complete.

Next, we assume that $a$, $b$ and $k$ satisfy one of the following cases:

Case 2.1. $k=\frac{N-\alpha}{N-ps}, a>0, 0<b<S_\alpha^{-k},$

Case 2.2. $k=\frac{2N-ps-\alpha}{2(N-ps)},a>0, b>0.$

Denote

and

Then, we have the following compactness result.

Lemma 2.2. Suppose that $a, b>0$ and $1<p<pk<p_\alpha^*$, then the functional $I_\lambda$ satisfies the $(PS)_{c_{\lambda}}$ condition for $c_\lambda<c^*=\Lambda-D\lambda^{\frac{p}{p-1}}$.

Proof. Let {${u_n}$}$\subset X_0$ be a $(PS)_{c_\lambda}$ sequence for

It follows form (2.2) and (2.5) that

This implies that {${u_n}$} is bounded in $X_0$. Up to a subsequence, still denote by $\{u_n\}$, there exists $u\in X_0$ such that

By using the concentration compactness principle (^[6], Lemma 4.5), there exist $u\in X_0$, two Borel regular measures $\sigma$ and $\nu$, $J$ denumerable, at most countable set $\{x_j\}_{j\in J}\subset\bar{\Omega}$, and non-negative numbers $\{\sigma_j\}_{j\in J}, \{\nu_j\}_{j\in J}\subset[0,\infty)$, for all $j\in J$, such that

as $n\rightarrow \infty$. Fix $\varepsilon>0$, let $\phi_{\varepsilon,j}(x)\in C_0^\infty(B(x_{j}, 2\varepsilon))$ be a smooth cut-off function centered at $x_j$ such that $0\leq\phi_{\varepsilon,j}\leq 1,$ $\|\nabla \phi_{\varepsilon,j}\|_\infty\leq\frac{C}{\varepsilon}$, and

Clear $\{\phi_{\varepsilon,j} u_n\}$ is bounded in $X_0$, it follows from $\langle I_\lambda'(u_n),\phi_{\varepsilon,j} u_n\rangle\rightarrow 0$ as $n\rightarrow \infty$ that

From the first term in (2.8), we have

Note that $\{u_n\}$ is bounded in $X_0$, the third term in (2.9), by using the H$\ddot{\mathrm{o}}$lder inequality and Lemma 2.3 in ^[22], we get

where $C>0$ is a positive constant. Letting $\varepsilon\rightarrow 0$, by (2.2) and (2.7), we get

Combining the above facts with (2.8), we have

Hence, taking the limit for $n\rightarrow \infty$ and $\varepsilon\rightarrow 0$ in (2.8), it follows from (2.10) and (2.11) that

This together with (2.7) implies that either $\nu_{j}=0$ or

From (2.7) and (2.12), we obtain that $\sigma_{j}=0$ or

To proceed further we show that (2.12) and (2.13) are impossible. Indeed, by contradiction, we assume that there exists $j_0\in J$ such that (2.12) and (2.13) hold. Applying (2.2), (2.7) and the Sobolev inequality, we get

By using the Young inequality, when $a>0$, we have

Consequently, we deduce that $c_\lambda\geq c^*=\Lambda-D\lambda^{\frac{p}{p-1}}$. This is a contradiction. Hence $\sigma_{j}=\nu_{j}=0$ for all $j\in J$, which implies that

as $n\rightarrow \infty$. Now, we prove that $u_n\rightarrow u$ in $X_0$, let $\varphi \in X_0$ be fixed and $B_{\varphi}$ be the linear functional on $X_0$ defined by

for every $v\in X_0$. By using the H$\ddot{\mathrm{o}}$lder inequality, one has

According to $I_\lambda'(u_n)\rightarrow 0$ in $X_0^{*}$ and $u_n\rightharpoonup u$ in $X_0$, we have

Since $\{u_n\}$ is bounded in $X_0$, by (2.6), one has

then

By $a, b>0$, we get

Moreover, it follows from the Brezis-Lieb Lemma that

This together with (2.14) and the H$\ddot{\mathrm{o}}$lder inequality, one has

Let us now recall the well-known Simon inequalities. That is, for every $\xi,\eta\in \mathbb{R}^N$

where $c_p, C_p>0$ depending only on $p$. According to (2.18), we distinguish two cases:

Case 2.3. if $p\geq 2$, it follows from (2.17) and (2.18) as $n\rightarrow \infty$ that

Case 2.4. if $1<p<2$, letting $\xi=u_n(x)-u_n(y)$ and $\eta=u(x)-u(y)$ in (2.18) as $n\rightarrow \infty$, we get

Indeed, since $\|u_n\|^p$ and $\|u\|^p$ are bounded in $X_0$, by the subadditivity inequality, for all $\xi,\eta\geq0$ and $1<p<2$, one has

Thus, we obtain that $u_n\rightarrow u$ in $X_0$. The proof is complete.

From ^[13], let $0\leq \alpha<ps<N$, for all minimizer $U_\alpha$ for $S_\alpha$, there exist $x_0\in \mathbb{R}^N$ and a non-increasing $u: \mathbb{R}^+\rightarrow \mathbb{R}$ such that $U_\alpha=u(|x-x_0|)$. Next, we fix a positive radially symmetric decreasing minimizer $U_\alpha=U_\alpha(r)$ for $S_\alpha$, multiplying $U_\alpha$ by a positive constant, we assume that

Then, we have the following Lemma.

Lemma 2.3. (^[13]) There exist $c_1, c_2>0$ and $\theta>1$ such tha

For any $\varepsilon>0$, the function

is also a minimizer for $S_\alpha$. For all $\delta\geq\varepsilon>0$, let

and

as well as

The functions $g_{\varepsilon,\delta}$ and $G_{\varepsilon,\delta}$ are nondecreasing and absolutely continuous. Consider now the radiallysymmetric non-increasing function $u_{\varepsilon,\delta}(r)=G_{\varepsilon,\delta}(U_{\alpha, \varepsilon}(r))$, which satisfies

Moreover, from ^[6,13], there exists $C>0$ such that for all $0<2\varepsilon\leq\delta<\theta^{-1}\delta_{\Omega}$, we have

Then we have the following Lemma.

Lemma 2.4. Suppose that $0\leq\alpha<ps<N$ and $\frac{N(p-2)+ps}{p-1}<\beta<\frac{N(p_\alpha^*-1)+\alpha}{p_\alpha^*}$.Then there exists $\Lambda_*>0$, for all $\lambda\in(0,\Lambda_*)$ such that $\sup_{t\geq 0} I_\lambda(tu_\varepsilon)<c^*=\Lambda-D\lambda^{\frac{p}{p-1}}$( where $c^*$ isthe constant given in Lemma 2.2).

Proof. By Lemma 2.1, we obtain that $I_\lambda(tu)\rightarrow -\infty$ as $t\rightarrow \infty$ and $I_\lambda(tu)<0$ as $t\rightarrow 0$, then there exists $t_{\varepsilon}>0$ such that $I_\lambda(t_{\varepsilon} u)=\sup_{t>0}I_\lambda(tu)\geq r>0$. Assume that there exist positive constants $t_1,t_2>0$ such that $0<t_1<t_{\varepsilon}<t_2<+\infty$. Without loss of generality, we take $\delta=1$ in the definition of $u_{\varepsilon,\delta}\in W^{s,p}(\Omega)$ given in Lemma 2.3, for any sufficiently small $0<\varepsilon<1$, set $u_\varepsilon=u_{\varepsilon,1}$ and $I_\lambda(tu_{\varepsilon})=J(t)-\lambda \int_{\Omega}\frac{u_\varepsilon}{|x|^\beta}dx$, where

It is easy to see that $\lim_{t\rightarrow0}J(t)=0$ and $\lim_{t\rightarrow \infty}J(t)=-\infty$. Hence, there exists $t_{\varepsilon}>0$ such that $J(t_{\varepsilon})=\max_{t\geq0}J(t)$, that is

By (2.20), we have

In addition, by the definition of $u_\varepsilon$, we have

where $\beta<\frac{N(p_\alpha^*-1)+\alpha}{p_\alpha^*}=N-\frac{N-ps}{p}$ and $C_1>0$ is a positive constant. Now, we consider the following two cases:

For Case 2.1, if $k=\frac{N-\alpha}{N-ps}, a>0, 0<b<S_\alpha^{-k}$ and $p_\alpha^*=pk$. It follows from (2.19), (2.20) and (2.22) that

Consequently, from the above information, we obtain

where $C_1, C_2>0$ (independent of $\varepsilon,\lambda$). Here we have used the fact that $\frac{N(p-2)+ps}{p-1}<\beta<\frac{N(p_\alpha^*-1)+\alpha}{p_\alpha^*}$, and let $\varepsilon=\lambda^{\frac{p}{N-ps}}$, $0<\lambda<\Lambda_1=\min\{1,(\frac{C_2+D}{C_1})^{\frac{(N-ps)(p-1)}{p[(N-\beta)(p-1)-(N-ps)]}}\;\;\;\}$,then

For Case 2.2, if $k=\frac{2N-ps-\alpha}{2(N-ps)},a>0, b>0.$ According to (2.19), (2.20) and (2.22), we have

where $C_3>0$ (independent of $\varepsilon,\lambda$). Consequently, it is similar to Case 2.1, by (2.23), there exists $\Lambda_2>0$ such that $0<\lambda<\Lambda_2$, we get

The proof is complete.

3.   Proof of Theorem 1.1

Theorem 3.1. Suppose that $0<\lambda<\Lambda_*$ $(\Lambda_*=\min\{\Lambda_0, \Lambda_1, \Lambda_2, 1\})$. Then the Eq (1.1) has two positive solutions.

Proof. It follows from Lemma 2.1 that

By the Ekland variational principle ^[8], there exists a minimizing sequence $\{u_n\}\subset \overline{B\rho(0)}$ such that

Hence, we obtain that $I_\lambda(u_n)\rightarrow m$ and $I_\lambda'(u_n)\rightarrow 0$ as $n\rightarrow \infty$. By Lemma 2.2, we have $u_n\rightarrow u_\lambda$ in $X_0$ with $I_\lambda(u_n)\rightarrow m<0$, which implies that $u_\lambda\not \equiv0$. Note that $I_\lambda(u_n)=I_\lambda(|u_n|)$, we have $u_\lambda\geq0$. Thus, by using the strong maximum principle (^[14], Lemma 2.3), we obtain that $u_\lambda$ is a positive solution of Eq (1.1) such that $I_\lambda(u_\lambda)<0$.

Applying the Mountain pass Lemma ^[2], Lemmas 2.1 and 2.2, there exists a sequence $\{u_n\}\subset X_0$ such that

where

and

According to Lemma 2.2, we know that $\{u_n\}\subset X_0$ has a convergent subsequence, still denoted by $\{u_n\}$, we may assume that $u_n\rightarrow u_*$ in $X_0$ as $n\rightarrow \infty$.

which implies that $u_*\not\equiv0$. Similarly, we can obtain that $u_*$ is a positive solution of Eq (1.1) with $I_\lambda(u_{*})>0$. That is, the proof of Theorem 1.1 is complete.

4.   Conclusions

In this paper, we consider a class of fractional $p$-Kirchhoff type equations with critical growth. Under some suitable assumptions, by using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of positive solutions.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 11661021). Supported by the projects of Education Department of Guizhou province (No. [2018]140). The school level fund project of Guizhou Minzu University (No. GZMUZK[2021]QN05).

Conflict of interest

The authors declare no conflict of interest in this paper.