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

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.

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

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,

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

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

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

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

we define

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

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

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:

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].

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

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 $R>0$,

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

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

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

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

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

(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

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

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

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

By a simple computation, it follows that

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

Hence, we have

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

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

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

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

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,

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

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

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

Combining with (18), we have

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

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. □

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).

Affiliations

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

   • Zeng-Qi Ou
   •  & Chun Li

Contributions

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

Corresponding author

Correspondence to Zeng-Qi Ou.

Competing interests

The authors declare that they have no competing interests.

Abbreviations

Not applicable.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Reprints and Permissions