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

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.

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:

  1. (i)

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

  2. (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].

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. □

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Acknowledgements

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

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Funding

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

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

Correspondence to Zeng-Qi Ou.

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MSC

  • 35D30
  • 35J50
  • 35J92

Keywords

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