Existence of nontrivial solutions for a class of nonlocal Kirchhoff type problems

Zeng-Qi Ou; Chun Li

doi:10.1186/s13661-018-1080-1

Research
Open Access

Existence of nontrivial solutions for a class of nonlocal Kirchhoff type problems

Zeng-Qi Ou¹Email author and
Chun Li¹

Boundary Value Problems20182018:158

https://doi.org/10.1186/s13661-018-1080-1

Received: 11 April 2018
Accepted: 11 October 2018
Published: 22 October 2018

Abstract

With the aid of the three-critical-point theorem due to Brezis and Nirenberg (see Brezis and Nirenberg in Commun. Pure Appl. Math. 44:939–963, 1991), two existence results of at least two nontrivial solutions for a class of nonlocal Kirchhoff type problems are obtained.

Keywords

Kirchhoff type problem
$(PS)$ condition
Local linking
Critical point

MSC

35D30
35J50
35J92

1 Introduction and main results

Consider the existence of weak solutions for the following nonlocal Kirchhoff type problem:

$$ \textstyle\begin{cases} - (a+b\int_{\Omega}|\nabla u|^{2}\,dx )\Delta u=f(x,u) &\mbox{in } \Omega, \\ u=0& \mbox{on } \partial\Omega, \end{cases} $$

(1)

where Ω is a smooth bounded domain in $R^{N}$ ($N\geq1$), $a>0$, $b>0$ are real numbers, and the nonlinearity $f\in C(\bar{\Omega}\times R, R)$.

Problem (1) is analogous to the stationary case of equations that arise in the study of string or membrane vibrations, that is,

$$u_{tt}-\biggl(a+b \int_{\Omega}|\nabla u|^{2}\,dx\biggr)\Delta u=f(x,u), $$

which was first proposed by Kirchhoff (see [3]) in 1883 to describe the transversal oscillations of a stretched string. Especially, in recent years, many solvability conditions with f (or F) near zero and infinity were considered to study the existence and multiplicity of weak solutions for problem (1) by using variational methods, for example, the nonlinearity f is asymptotically 3-linear at infinity (see [4, 6, 9]), the nonlinearity f is 3-suplinear at infinity (see [5, 7, 9]), and the nonlinearity f is 3-sublinear at infinity (see [9]). In this paper, motivated by [2, 7, 8], we prove the existence of at least two nontrivial solutions for problem (1) by using the variational method.

Let $H_{0}^{1}(\Omega)$ be the usual Hilbert space with the norm

$$\|u\|= \biggl( \int_{\Omega}|\nabla u|^{2} \,dx \biggr)^{\frac{1}{2}} \quad \mbox{for any } u\in H_{0}^{1}(\Omega). $$

From the Rellich embedding theorem, the embedding $H_{0}^{1}(\Omega )\hookrightarrow L^{\theta}(\Omega)$ is continuous for any $\theta\in [1,2^{*}]$ and compact for any $\theta\in[1,2^{*})$, where $2^{*}=+\infty$ if $N=1,2$ and $2^{*}=\frac{2N}{N-2}$ if $N\geq3$. Moreover, for any $\theta\in[1,2^{*})$, there is a constant $\tau_{\theta}>0$ such that

$$ \|u\|_{L^{\theta}}\leq\tau_{\theta}\|u\|\quad \mbox{for any } u\in H_{0}^{1}(\Omega), $$

(2)

where $\|\cdot\|_{L^{\theta}}$ denotes the norm of $L^{\theta}(\Omega)$. Let $m(x)\in C(\bar{\Omega})$ be positive on a subset of positive measure, the following eigenvalue problem

$$ \textstyle\begin{cases} -\Delta u=\lambda m(x)u &\mbox{in } \Omega, \\ u=0& \mbox{on } \partial\Omega \end{cases} $$

(3)

has a sequence of variational eigenvalues $\{\lambda_{k}(m)\}$ such that $\lambda_{1}(m)<\lambda_{2}(m)<\cdots<\lambda_{k}(m)\to\infty$ as $k\to \infty$. Let $M(x)\in C(\bar{\Omega})$ be positive on Ω. For the following nonlinear eigenvalue problem

$$ \textstyle\begin{cases} -\|u\|^{2}\Delta u=\mu M(x)u^{3} &\mbox{in } \Omega, \\ u=0& \mbox{on } \partial\Omega, \end{cases} $$

(4)

we define

$$\mu_{1}(M)=\inf\biggl\{ \|u\|^{4}:u\in H_{0}^{1}( \Omega), \int_{\Omega}M(x)u^{4}\,dx=1\biggr\} . $$

Similar to Lemma 2.1 of [9], we can prove that $\mu_{1}(M)$ is the first eigenvalue of (4) and positive. Moreover, there is an eigenvalue $\Phi_{1}^{M}$ such that $\Phi_{1}^{M}>0$ in Ω.

Let $m_{0}(x)\in C(\bar{\Omega})$ be positive on a subset of positive measure and $m_{\infty}(x)\in C(\bar{\Omega})$ be positive on Ω. Assume that

$$\begin{aligned}& \lim_{|t|\to0}\frac{2F(x,t)}{at^{2}}=m_{0}(x) \quad \mbox{uniformly in }x\in \Omega, \end{aligned}$$

(5)

$$\begin{aligned}& \lim_{|t|\to\infty}\frac{4F(x,t)}{bt^{4}}=m_{\infty}(x) \quad \mbox{uniformly in }x\in\Omega, \end{aligned}$$

(6)

$$\begin{aligned}& \lim_{|t|\to\infty}\bigl(f(x,t)t-4F(x,t)\bigr)=+\infty \quad \mbox{uniformly in }x\in\Omega, \end{aligned}$$

(7)

where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$. We are ready to state our main results.

Theorem 1

Let $N=1,2,3$, and assume that the function F satisfies (5) with $\lambda_{k}(m_{0})<1<\lambda_{k+1}(m_{0})$ for some $k\geq1$ and (6), and there exist $4< p<2^{*}$ and $c_{0}>0$ such that

$$ \bigl\vert f(x,t) \bigr\vert \leq c_{0}\bigl(1+ \vert t \vert ^{p-1}\bigr) \quad \textit{for any }(x,t)\in\bar{\Omega }\times R, $$

(8)

then problem (1) has at least two nontrivial solutions in each of the following cases:

(i)

$\mu_{1}(m_{\infty})>1$ or
(ii)

$\mu_{1}(m_{\infty})=1$ and (7) hold.

Theorem 2

Assume that the nonlinearity F satisfies (5) with $\lambda_{k}(m_{0})<1<\lambda_{k+1}(m_{0})$ for some $k\geq1$ and the following condition:

$$ \lim_{|t|\to\infty}\frac{f(x,t)}{|t|^{p-1}}=0\quad \textit{uniformly in }x\in \Omega, $$

(9)

where $p=4$ if $N=1,2,3$ and $p=2^{*}$ if $N\geq4$, then problem (1) has at least two nontrivial solutions.

Remark

If $N=1, 2, 3$ and the nonlinearity f is 3-suplinear at infinity, Sun and Tang in [7] obtained a nontrivial solution for problem (1) by using the local linking theorem due to Li and Willem. In [8], when the nonlinearity F is some asymptotically 4-linear at infinity, Yang and Zhang proved the existence of at least two nontrivial solutions for problem (1) by means of the Morse theory and local linking. Since $p=2^{*}\leq4$ ($N\geq4$), condition (9) implies that the nonlinearity f is 3-sublinear at infinity. Hence, our results are the complements for the ones of [7, 8].

2 Proof of the theorems

Define the functional $I: H_{0}^{1}(\Omega)\to R$ as follows:

$$ I(u)=\frac{b}{4}\|u\|^{4}+\frac{a}{2}\|u \|^{2}- \int_{\Omega}F(x,u)\,dx. $$

(10)

From (8) (or (9)), by a standard argument, the functional $I\in C^{1}( H_{0}^{1}(\Omega), R)$, and a weak solution of problem (1) is a critical point of the functional I in $H_{0}^{1}(\Omega)$.

Recall that a sequence $\{u_{n}\}\subset H_{0}^{1}(\Omega)$ is called a $(PS)_{c}$ sequence for any $c\in R$ of the functional I on $H_{0}^{1}(\Omega)$ if $I(u_{n})\to c$ and $I'(u_{n})\to0$ as $n\to\infty$. The functional I is called to satisfy the $(PS)_{c}$ condition if any $(PS)_{c}$ sequence has a convergent subsequence. We will prove our theorems by using the following three-critical-point theorem related to local linking due to Brezis and Nirenberg (see Theorem 4 in [1]).

Theorem A

Let X be a Banach space with a direct sum decomposition $X=X_{1}\oplus X_{2}$ with $\dim X_{1}<\infty$. Let I be a $C^{1}$ function on X with $I(0)=0$ satisfying the $(PS)$ condition, and assume that, for some $R>0$,

$$\textstyle\begin{cases} I(u)\leq0 \quad \textit{for }u\in X_{1}, \|u\|\leq R, \\ I(u)\geq0 \quad \textit{for }u\in X_{2}, \|u\|\leq R. \end{cases} $$

Assume also that I is bounded below and $\inf_{X} I<0$. Then I has at least two nonzero critical points.

Proof of Theorem 1

(a) The functional I satisfies the local linking at zero with respect to $(V_{k}, V_{k}^{\bot})$, where $V_{k}=\bigoplus_{i=1}^{k}\ker(-\Delta-\lambda_{i}(m_{0}))$ and $V_{k}^{\bot}=\bigoplus _{i=k+1}^{+\infty}\ker(-\Delta-\lambda_{i}(m_{0}))$ such that $H_{0}^{1}(\Omega )=V_{k}\oplus V_{k}^{\bot}$.

In fact, from (5), for any $\varepsilon>0$, there is a positive constant $L_{0}$ such that

$$\bigl\vert 2F(x,t)-am_{0}(x)t^{2} \bigr\vert \leq a \varepsilon t^{2} \quad \mbox{for any }x\in\Omega \mbox{ and } \vert t \vert \leq L_{0}. $$

Combining the continuity of F, (8), and the above inequality, there is $M_{0}=M_{0}(\varepsilon)>0$ such that

$$\begin{aligned}& F(x,t)\geq\frac{a}{2}m_{0}(x)t^{2}- \frac{a\varepsilon}{2}t^{2}-M_{0}|t|^{p}\quad \mbox{for any } (x,t)\in\Omega\times R,\quad \mbox{and} \end{aligned}$$

(11)

$$\begin{aligned}& F(x,t)\leq\frac{a}{2}m_{0}(x)t^{2}+ \frac{a\varepsilon}{2}t^{2}+M_{0}|t|^{p} \quad \mbox{for any } (x,t)\in\Omega\times R. \end{aligned}$$

(12)

For any $u\in V_{k}$, from (2), (10), and (11), it follows that

$$\begin{aligned} I(u) \leq&\frac{b}{4}\|u\|^{4}+\frac{a}{2}\|u \|^{2}-\frac{a}{2} \int_{\Omega}m_{0}(x)|u|^{2}\,dx+ \frac{a\varepsilon}{2} \int_{\Omega}|u|^{2}\,dx+M_{0} \int_{\Omega}|u|^{p}\,dx \\ \leq&\frac{a}{2} \biggl(1-\frac{1}{\lambda_{k}(m_{0})}+\varepsilon\tau _{2}^{2} \biggr)\|u\|^{2}+\frac{b}{4}\|u \|^{4}+M_{0} \tau_{p}^{p}\|u \|^{p}. \end{aligned}$$

(13)

On the other hand, for any $u\in V_{k}^{\bot}$, from (2), (10), and (12), we obtain

$$\begin{aligned} I(u) \geq&\frac{b}{4}\|u\|^{4}+\frac{a}{2}\|u \|^{2}-\frac{a}{2} \int_{\Omega}m_{0}(x)|u|^{2}\,dx- \frac{a\varepsilon}{2} \int_{\Omega}|u|^{p}\,dx-M_{0} \int_{\Omega}|u|^{p}\,dx \\ \geq&\frac{a}{2} \biggl(1-\frac{1}{\lambda_{k+1}(m_{0})}-\varepsilon\tau ^{2}_{2} \biggr)\|u\|^{2}+\frac{b}{4}\|u \|^{4}-M_{0} \tau_{p}^{p}\|u \|^{p}. \end{aligned}$$

(14)

Noting that $\lambda_{k}(m_{0})<1<\lambda_{k+1}(m_{0})$ and $4< p<2^{*}$, (13) and (14), let $\varepsilon=\min\{(1-\lambda _{k}(m_{0}))/\lambda_{k}(m_{0}), (\lambda_{k+1}(m_{0})-1)/\lambda_{k+1}(m_{0})\}/2\tau^{2}_{2}$, there is a constant $r_{0}>0$ such that

$$\begin{aligned}& I(u)< 0\quad \mbox{for any } u\in V_{k}\mbox{ with }0< \|u\|\leq r_{0}, \\& I(u)> 0\quad \mbox{for any } u\in V_{k}^{\bot}\mbox{ with } 0< \|u\|\leq r_{0}. \end{aligned}$$

(b) The functional I satisfies the $(PS)$ condition. To the end, it suffices to say the functional I is coercive on $H^{1}_{0}(\Omega)$, i.e., $I(u)\to+\infty$ as $\|u\|\to\infty$.

If $\mu_{1}(m_{\infty})>1$, by (6), for any $\varepsilon>0$, there is $L_{1}>0$ such that

$$\bigl\vert 4F(x,t)-b m_{\infty}(x)t^{4} \bigr\vert \leq b \varepsilon t^{4} \quad \mbox{for any }x\in \Omega\mbox{ and } \vert t \vert \geq L_{1}. $$

Hence, from the continuity of F, there exists $M_{1}=M_{1}(\varepsilon )>0$ such that

$$ F(x,t)\leq\frac{b}{4}m_{\infty}(x)t^{4}+ \frac{b\varepsilon}{4}t^{4}+M_{1}\quad \mbox{for any } (x,t)\in \Omega\times R. $$

(15)

From (2), (10), and (15), we obtain

$$\begin{aligned} I(u) \geq&\frac{b}{4}\|u\|^{4}+\frac{a}{2}\|u \|^{2}-\frac{b}{4} \int_{\Omega }m_{\infty}(x)|u|^{4}\,dx - \frac{b\varepsilon}{4} \int_{\Omega}|u|^{4}\,dx-M_{1}|\Omega| \\ \geq&\frac{b}{4} \biggl(1-\frac{1}{\mu_{1}(m_{\infty})} -\varepsilon \tau_{4}^{4} \biggr)\|u\|^{4}-M_{1}| \Omega|, \end{aligned}$$

where $|\Omega|$ denotes the Lebesgue measure of Ω. Hence, for $\varepsilon>0$ small enough, it follows that the functional I is coercive on $H^{1}_{0}(\Omega)$.

If $\mu_{1}(m_{\infty})=1$ and (7) hold, let

$$H(x,t)=F(x,t)-\frac{b}{4}m_{\infty}(x)t^{4}. $$

By a simple computation, it follows that

$$H'(x,t)t-4H(x,t)=f(x,t)t-4F(x,t). $$

From (7), for any $M_{2}>0$, there is $L_{2}>0$ such that

$$H'(x,t)t-4H(x,t)\geq M_{2}\quad \mbox{for any }x\in \Omega\mbox{ and }|t|\geq L_{2}. $$

Hence, we have

$$\frac{d}{ds} \biggl(\frac{H(x,s)}{s^{4}} \biggr)= \frac{H'(x,s)s-4H(x,s)}{s^{5}}\geq \frac{M_{2}}{s^{5}}\quad \mbox{for any }x\in \Omega\mbox{ and }|s|\geq L_{2}. $$

Integrating the above expression over the interval $[t,T]\subset [L_{2},\infty)$, we obtain

$$\frac{H(x,t)}{t^{4}}\leq\frac{H(x,T)}{T^{4}}+\frac{M_{2}}{4} \biggl( \frac {1}{T^{4}}-\frac{1}{t^{4}} \biggr). $$

Noting that $\lim_{|T|\to\infty}H(x,T)/T^{4}=0$, let $T\to+\infty$, we obtain $H(x,t)\leq-M_{2}/4$ for $t\geq L_{2}$ and $x\in\Omega$. Similarly, $H(x,t)\leq-M_{2}/4$ for $t\leq-L_{2}$ and $x\in\Omega$. Hence, from the arbitrariness of $M_{2}(>0)$, we have

$$\lim_{|t|\to\infty}H(x,t)=-\infty\quad \mbox{uniformly in }x\in\Omega. $$

Moreover, from the continuity of F, there is a positive constant $M_{3}$ such that

$$ H(x,t)< M_{3}\quad \mbox{for any } (x,t)\in\Omega\times R. $$

(16)

If the functional I is not coercive on $H^{1}_{0}(\Omega)$, there are a sequence $\{u_{n}\}\subset H^{1}_{0}(\Omega)$ and a positive constant $M_{4}$ such that $\|u_{n}\|\to\infty$ as $n\to\infty$ and $I(u_{n})\leq M_{4}$. By the definition of $\mu_{1}(m_{\infty})$ and $\mu_{1}(m_{\infty})=1$, we have that $\int_{\Omega}m_{\infty}(x)|u_{n}|^{4}\,dx\leq\|u_{n}\|^{4}$. Hence, from (16), it follows that

$$\begin{aligned} M_{4}\geq I(u_{n}) =&\frac{b}{4}\|u_{n} \|^{4}+\frac{a}{2}\|u_{n}\|^{2}- \frac {b}{4} \int_{\Omega}m_{\infty}(x)|u_{n}|^{4} \,dx- \int_{\Omega}H(x,u_{n})\,dx \\ \geq&\frac{b}{4}\|u_{n}\|^{4}+\frac{a}{2} \|u_{n}\|^{2}-\frac{b}{4} \int_{\Omega}m_{\infty}(x)|u_{n}|^{4} \,dx-M_{3}|\Omega| \\ \geq&\frac{a}{2}\|u_{n}\|^{2}-M_{3}| \Omega| \\ \to& +\infty \quad \mbox{as } n\to\infty, \end{aligned}$$

which is a contradiction, and the conclusion is proved.

(c) From (b), we have that the functional I is bounded from below. From the fact that $I(u)<0$ for any $u\in V_{k}$ with $0<\|u\|\leq r_{0}$, we have $\inf_{u\in H^{1}_{0}(\Omega)}I(u)<0$. Moreover, $I(0)=0$. Therefore, Theorem 1 is proved by Theorem A. □

Proof of Theorem 2

First of all, from (a) of the proof of Theorem 1, we have that the functional I satisfies the local linking at zero with respect to $(V_{k}, V_{k}^{\bot})$. And then, we know from (9) that $f(x,t)$ is 3-sublinear at infinity, which implies that the functional I is coercive on $H^{1}_{0}(\Omega)$ by a standard argument. We obtain that the functional I is bounded from below and satisfies the $(PS)$ condition for $N=1,2,3$. In the following, we only prove that the functional I also satisfies the $(PS)$ condition for $p=2^{*}$ ($N\geq4$), where $f(x,t)$ is not only 3-sublinear at infinity, but also is asymptotically critical growth at infinity.

In fact, let $\{u_{n}\}$ be a $(PS)$ sequence of I, that is,

$$ I(u_{n})\to c,\qquad I'(u_{n}) \to0\quad \mbox{as }n\to\infty. $$

(17)

Noting that the functional I is coercive on $H^{1}_{0}(\Omega)$, we obtain that $\{u_{n}\}$ is bounded in $H^{1}_{0}(\Omega)$. Going if necessary to a subsequence, we can assume $u_{n}\rightharpoonup u$ in $H^{1}_{0}(\Omega )$, and by the Rellich theorem, $u_{n}\to u$ in $L^{r}(\Omega)$ ($1\leq r<2^{*}$). From (17) and the boundedness of $\{u_{n}\}$, we have

$$ \bigl\langle I'(u_{n}),u_{n}-u \bigr\rangle =\bigl(a+b\|u_{n}\|^{2}\bigr) \int_{\Omega}\nabla u_{n}(\nabla u_{n}-\nabla u)\,dx+ \int_{\Omega}f(x,u_{n}) (u_{n}-u)\,dx\to0 $$

(18)

as $n\to\infty$. From (9), for any $\varepsilon>0$, there is $M_{5}>0$ such that

$$\bigl\vert f(x,t) \bigr\vert \leq\varepsilon|t|^{p-1}+M_{5} \quad \mbox{for any }(x,t)\in\Omega \times R. $$

Hence, from Hölder’s inequality, (2), the boundedness of $\{ u_{n}\}$, and the arbitrariness of ε, we have

$$\begin{aligned} \biggl\vert \int_{\Omega}f(x,u_{n}) (u_{n}-u)\,dx \biggr\vert \leq& \int_{\Omega}\bigl(\varepsilon|u_{n}|^{p-1}+M_{5} \bigr)|u_{n}-u|\,dx \\ \leq&\varepsilon \int_{\Omega}\bigl(|u_{n}|^{p}+|u_{n}|^{p-1}|u| \bigr)\,dx+M_{5} \Vert u_{n}-u \Vert _{L^{1}} \\ \leq&\varepsilon \Vert u_{n} \Vert _{L^{p}}^{p-1} \bigl( \Vert u_{n} \Vert _{L^{p}}+ \Vert u \Vert _{L^{p}}\bigr)+M_{5} \Vert u_{n}-u \Vert _{L^{1}} \\ \to& 0 \quad \mbox{as }n\to\infty. \end{aligned}$$

Combining with (18), we have

$$\int_{\Omega}\nabla u_{n}(\nabla u_{n}-\nabla u)\,dx\to0 \quad \mbox{as }n\to\infty. $$

Since $u_{n}\rightharpoonup u$ weakly in $H^{1}_{0}(\Omega)$, we have

$$\int_{\Omega}\nabla u(\nabla u_{n}-\nabla u)\,dx\to0 \quad \mbox{as } n\to\infty. $$

Then $u_{n}\rightarrow u$ strongly in $H^{1}_{0}(\Omega)$ as $n\rightarrow \infty$.

At last, similar to (c) of the proof of Theorem 1, Theorem 2 is proved. □

Declarations

Acknowledgements

The authors would like to thank the referees for their useful suggestions.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Funding

The work is supported by the National Natural Science Foundation of China (No. 11471267).

Authors’ contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)

School of Mathematics and Statistics, Southwest University, Chongqing, People’s Republic of China

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