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Existence of nontrivial solutions for a class of nonlocal Kirchhoff type problems
Boundary Value Problemsvolume 2018, Article number: 158 (2018)
Abstract
With the aid of the threecriticalpoint 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:
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 3linear at infinity (see [4, 6, 9]), the nonlinearity f is 3suplinear at infinity (see [5, 7, 9]), and the nonlinearity f is 3sublinear 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 $2^{*}=\frac{2N}{N2}$ 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:

(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:
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 3suplinear 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 4linear 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 3sublinear 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:
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 threecriticalpoint 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 $H(x,t)\leqM_{2}/4$ for $t\geq L_{2}$ and $x\in\Omega$. Similarly, $H(x,t)\leqM_{2}/4$ for $t\leqL_{2}$ 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 3sublinear 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 3sublinear 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. □
References
 1.
Brezis, H., Nirenberg, L.: Remarks on finding critical points. Commun. Pure Appl. Math. 44, 939–963 (1991)
 2.
Furtado, M.F., de Paiva, F.O.V.: Multiplicity of solutions for resonant elliptic systems. J. Math. Anal. Appl. 319, 435–449 (2006)
 3.
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
 4.
Liang, Z.P., Li, F.Y., Shi, J.P.: Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 31, 155–167 (2014)
 5.
Mao, A.M., Zhang, Z.T.: Signchanging and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 70, 1275–1287 (2009)
 6.
Perera, K., Zhang, Z.T.: Nontrivial solutions of Kirchhofftype problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)
 7.
Sun, J.J., Tang, C.L.: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal. 74, 1212–1222 (2011)
 8.
Yang, Y., Zhang, J.H.: Nontrivial solutions of a class of nonlocal problems via local linking theory. Appl. Math. Lett. 23, 377–380 (2010)
 9.
Zhang, Z.T., Perera, K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456–463 (2006)
Acknowledgements
The authors would like to thank the referees for their useful suggestions.
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The work is supported by the National Natural Science Foundation of China (No. 11471267).
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Correspondence to ZengQi Ou.
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MSC
 35D30
 35J50
 35J92
Keywords
 Kirchhoff type problem
 $(PS)$ condition
 Local linking
 Critical point