# 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.

## 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 ) 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 ). 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 , 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  obtained a nontrivial solution for problem (1) by using the local linking theorem due to Li and Willem. In , 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 ).

### 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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Correspondence to Zeng-Qi Ou.

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