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Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations
Boundary Value Problems volume 2021, Article number: 38 (2021)
Abstract
In this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.
1 Introduction
In recent ten years, research on the existence numbers of solutions to nonlinear differential equations has been widely performed via corresponding critical point theorems, for example, [2, 6, 9–11, 14–17] and the references therein, and the aim of this paper is to obtain the existence of two solutions to Kirchhoff-type fourth-order impulsive elastic beam equations.
In [5], Bonanno et al. considered the existence of two positive solutions for superlinear Neumann problems with a complete Sturm–Liouville operator:
under the Amberosetti–Rabinowitz condition combining with a local condition not adding on zero point.
In [8], D’Aguì et al. studied the existence results of two non-zero solutions for some Sturm–Liouville equations involving the p-Laplacian operators with Robin boundary conditions:
In [13], we obtained the existence of triple solutions of the following second-order Hamiltonian systems with impulsive effects:
via variational methods and three-critical-point theorems without adding superlinear or sublinear assumptions to the nonlinearity at zero nor infinity.
In [12], we obtained the existence of at least three solutions to the impulsive equations with small non-autonomous perturbations
when the nonlinearity satisfies some superlinear conditions.
In [7], D’Aguì et al. considered the fourth-order differential equations with impulsive effects
where \(A, B \) are real constants, \(f: [0,1] \times \mathbb{R} \to \mathbb{R}\) is a \(L^{1}\)-Carathéodory function. They gave some criteria to guarantee the differential equations have at least two non-trivial solutions.
Motivated by the above-mentioned work, in this article, we consider the following Kirchhoff-type fourth-order differential equations with impulsive effects:
where \(K: [0, +\infty )\to \mathbb{R}\) is a continuous function such that there exist two constants \(m_{0}\) and \(m_{1}\) satisfying \(0< m_{0}\leq K(x)\leq m_{1}, \forall x\geq 0\), \(\alpha \leq 0, \beta \geq 0\) are real constants, λ is a positive real parameter, \(f: [0,1] \times \mathbb{R} \to \mathbb{R}\) is a \(L^{1}\)-Carathéodory function, \(0=t_{0}< t_{1}<\cdots <t_{l}<t_{l+1}=1\), \(\Delta u''(t_{i})= u''(t_{i}^{+})- u''(t_{i}^{-}),\Delta u'''(t_{i})= u'''(t_{i}^{+})- u'''(t_{i}^{-})\), and \(I_{ji}\ (j=1,2; i=1,2,\ldots,l)\in C(\mathbb{R},\mathbb{R})\). We aim to get the existence of at least two solutions. We find the existence of at least two solutions without assuming any asymptotic conditions neither at zero or at infinity on nonlinear items. Our main tools are variational methods and two-critical-point theorems by Bonanno and Marano.
2 Preliminaries
We consider the spaces
Take \(X=H_{0}^{1}(0,1)\cap H^{2}(0,1)\), thus X is a Hilbert space with the inner product
and the induced norm
By direct calculation one finds that the norm \(\|u\|\) is equivalent to the following norm:
for more details, see [4].
It is well known that the embedding \(X\hookrightarrow C^{1}([0,1])\) is compact and there exists a positive constant \(k=1+\frac{1}{\pi }\) such that
for all \(u\in X\) (see [18]).
We call that \(u\in X\) is a weak solution of problem (1.6) if
holds for any \(v\in X\).
Put
Let the functional \(I_{\lambda }:X\rightarrow \mathbb{R}\) be defined by
where
Lemma 2.1
\(\Phi, \Psi \) are well defined and Gâteaux differentiable at any \(u\in X\) and
Lemma 2.2
If \(u\in X\) satisfying \(I_{\lambda }'(u)=0\), then u is a weak solution of the Kirchhoff-type system (1.6).
Theorem 2.3
([1], Theorem 3.2)
Let X be a real Banach space and let \(\Phi,\Psi: X \to R\) be two continuous Gâteaux differentiable functions such that Φ is bounded from below and \(\Phi (0)-\Psi (0)=0\). Fix \(r>0\) such that \({\sup_{\Phi (u)< r}\Psi (u)}<+\infty \) and assume that for each
the functional \(\Phi -\lambda \Psi \) satisfies the (P.S.)-condition and it is unbounded from below. Then, for each \(\lambda \in (0,\frac{r}{\sup_{\Phi (u)\leq r}\Psi (u)})\), the functional \(\Phi -\lambda \Psi \) admits at least two distinct critical points.
Theorem 2.4
([3], Theorem 2.1)
Let X be a real Banach space and let \(\Phi,\Psi: X \to R\) be two continuous Gâteaux differentiable functions such that \(\inf_{X}\Phi =\Phi (0)-\Psi (0)=0\). Assume there are \(r\in \mathbb{R}\) and \(\bar{u}\in X\), with \(0<\Phi (\bar{u})<r\) such that
and, for each \(\lambda \in ( \frac{\Phi (\bar{u})}{\Psi (\bar{u})}, \frac{r}{\sup_{\Phi (u)\leq r}\Psi (u)} )\), the functional \(I_{\lambda }=\Phi -\lambda \Psi \) satisfies the (P.S.)-condition and it is unbounded from below. Then, for each \(\lambda \in ( \frac{\Phi (\bar{u})}{\Psi (\bar{u})}, \frac{r}{\sup_{\Phi (x)\leq r}\Psi (u)} )\), the functional \(I_{\lambda }\) admits at least two non-zero critical points \(u_{\lambda,1},u_{\lambda,2}\) such that \(I_{\lambda }(u_{\lambda,1})<0<I_{\lambda }(u_{\lambda,2})\).
Definition 2.5
([7], Definition 2.1)
\(f:[0,1]\times \mathbb{R} \to \mathbb{R}\) is an \(L^{1}\)-Carathéodory function if:
-
(1)
\(x\mapsto f(x,\xi )\) is measurable for every \(\xi \in \mathbb{R}\);
-
(2)
\(\xi \mapsto f(x,\xi )\) is continuous for almost every \(x \in [0,1]\);
-
(3)
for every \(s>0\) there is a function \(l_{s} \in L^{1}([0,1])\) such that
$$ \sup_{ \vert \xi \vert \leq s} \bigl\vert f(x, \xi ) \bigr\vert \leq l_{s}(x) $$for a.e. \(x\in [0,1]\).
3 Main results
Our main results are the following two theorems about the existence of at least two distinct solutions to the Kirchhoff-type system (1.6).
Theorem 3.1
Assume the following assumptions:
(H1) there exists \(\mu > \frac{2m_{1}}{m_{0}}\) such that
(H2) there exist positive constants \(L_{i}(i=1,2,\ldots,l)\) such that
(H3) \(0< I_{1i}(u)u\leq \mu \int _{0}^{u'}I_{1i}(t)\,dt, 0< I_{2i}(u)u \leq \mu \int _{0}^{u}I_{2i}(t)\,dt\leq \mu \delta |u|^{2}, i=1,2,\ldots,l, \forall u\in \mathbb{R}\setminus \{0\}\), then there exists \(c>0\) such that, when
the functional \(I_{\lambda }\) admits at least two distinct critical points.
Proof
In view of condition (H3) and \(0< m_{0}\leq K(x)\leq m_{1}, \forall x\geq 0\), there exists a constant \(c_{1}> 0\), such that
thus one finds that Φ is bounded from below.
Let \(\{u_{n}\}\in X\) such that \(\{I_{\lambda }(u_{n})\}\) is bounded and \({I_{\lambda }'(u_{n})}\rightarrow 0 \), as \(n\rightarrow +\infty \), then we prove that \(\{u_{n}\}\) is bounded in X. In fact, combining (H1), (H3), (2.2) with (2.3), one has
which implies that \(\{u_{n}\}\) is bounded in view of \(\mu > \frac{2m_{1}}{m_{0}}\).
Hence there exists a subsequence \(\{u_{n_{k}}\}\) of \(\{u_{n}\}\) converging uniformly to u in \([0, 1]\). Thus, when \(k\rightarrow +\infty \), one has
Thus, by standard direct calculations, one finds that there exists a positive constant \(c_{2}\) satisfying \(a< c_{2}< b\), where
such that
and
Combining (3.1), (3.2) with (3.3), we get \(\|u_{n_{k}}-u\|_{X}\rightarrow 0\), as \(k\rightarrow +\infty \), thus \(\{u_{n_{k}}\}\) converges strongly to u in X, then the functional \(I_{\lambda }\) satisfies the (P.S.)-condition.
Next, we prove \(I_{\lambda }\) is unbounded from below.
Noticing that \(0<\mu F(t,u)\leq f(t,u)u,\forall t\in [0,1], u\in \mathbb{R} \setminus \{0\}\), one finds that there exist \(\bar{\alpha },\bar{\beta }>0\) such that \(F(t,u)\geq \bar{\alpha }|u|^{\mu }-\bar{\beta }, u \in \mathbb{R} \setminus \{0\}\). We choose \(\rho _{n}(t)=\xi _{n}\in \mathbb{R}\) satisfying \(| \xi _{n}|\rightarrow +\infty \), thus \(\xi _{n}\in X\). In view of (2.2) and \(\int _{0}^{u}I_{2i}(t)\,dt\leq \delta |u|^{2},i=1,2,\ldots,l,\forall u\in \mathbb{R}\setminus \{0\}\), we get
noticing that \(\mu >2\), and it leads to \(I_{\lambda }(\rho _{n})\rightarrow -\infty (|\rho _{n}|\rightarrow + \infty )\), thus one finds that the functional \(I_{\lambda }\) is unbounded from below.
Taking account of (2.1), (2.2) and (H3), for all \(u\in X\) satisfying \(\Phi (u) \leq r\), one has \(\|u\|\leq \sqrt{2r}\) because of \(\|u\|_{\infty } \leq k\|u\|_{H}\leq k\sqrt{2r}=:c\). Therefore,
Hence, by Theorem 2.3, one finds that the functional \(I_{\lambda }\) admits at least two distinct critical points for \(\lambda \in (0, \frac{c^{2}}{2k^{2}\int _{0}^{1}\max_{|u|\leq c}F(t,u)\,dx})\). □
Theorem 3.2
Assume the conditions of Theorem 3.1are satisfied. In addition, suppose there exist \(c>0\) and \(\bar{\xi }\in \mathbb{R}\) with \(|\bar{\xi }|<\frac{c}{k\sqrt{\beta m_{1}+2\delta l}}\), such that
thus when \(\lambda \in ( \frac{(\frac{1}{2}\beta m_{1}+\delta l)|\bar{\xi }|^{2}}{\int _{0}^{1}F(t,\bar{\xi })\,dx}, \frac{c^{2}}{2k^{2}\int _{0}^{1}\max_{|u|\leq c}F(t,u)\,dt})\), the Kirchhoff-type system (1.6) has at least two non-trivial solutions.
Proof
Choose \(\bar{v}(t)=\bar{\xi }\), taking account of condition (H3), one has
and
In view of (3.5) and (3.6), we get \(0<\Phi (\bar{v})<r\).
Noting that \(\Psi (\bar{v})= \int _{0}^{1}F(t,\bar{\xi })\,dt\), by virtue of (3.4), we get
so the conditions in Theorem 2.4 are all satisfied. Hence, we complete the proof. □
Remark 3.3
Noting that \(F(t,u)={ \int _{0}^{u}f(t,\xi )\,d\xi } \text{ for all }(t,u) \in [0,1] \times \mathbb{R}\) is continuous on \(u\in \mathbb{R}\), thus condition (H1) can be replaced by the following condition:
(H0) Suppose that there exist \(\mu >2\) and \(L>0\) such that
Example 3.4
Taking
By calculation we know that \(f(t,u)\) is a \(L^{1}\)-Carathéodory function in view of \(\int _{0}^{1}\frac{1}{\sqrt{t(1-t)}}\,dt\doteq 3.142\). Consider the Kirchhoff-type system (1.6), and we choose \(c=10, \delta =\frac{7}{8},\bar{\xi }=0.9\). There exists \(0<|\bar{u}|<1\) such that \(\max_{|u|\leq 10}F(t,u)=\max_{|u|\leq 1}F(t,u)=F(t,\bar{u})\), furthermore, by a Newton-iterative method we obtain \(\bar{u}=0.706, F(1,\bar{u})=24.88\), by calculation, the Kirchhoff-type system (1.6) has at least two non-trivial solutions when \(\lambda \in (0.029,0.369)\) by applying Theorem 3.2 and Remark 3.3.
4 Conclusion
The main novelty of our paper is that we apply a recent obtained critical-point theorem to the study of the superlinear fourth-order impulsive elastic beam equations, and the existence of at least two solutions of this kind of equations has been studied. The assumptions made and the related considerations are needed to set up the problem in a way that makes it suitable for the abstract framework, and we also improve many previous results.
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Funding
This work is supported by National Natural Science Foundation of China (11571197) and the Taishan Scholars Program of Shandong Province (tsqn20161041).
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Liu, J., Yu, W. Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations. Bound Value Probl 2021, 38 (2021). https://doi.org/10.1186/s13661-021-01515-8
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DOI: https://doi.org/10.1186/s13661-021-01515-8