- Research
- Open Access
- Published:
Existence and multiplicity of weak solutions for a nonlinear impulsive \((q,p)\)-Laplacian dynamical system
Advances in Difference Equations volume 2017, Article number: 128 (2017)
Abstract
In this paper, we investigate the existence and multiplicity of nontrivial weak solutions for a class of nonlinear impulsive \((q,p)\)-Laplacian dynamical systems. The key contributions of this paper lie in (i) Exploiting the least action principle, we deduce that the system we are interested in has at least one weak solution if the potential function has sub-\((q,p)\) growth or \((q,p)\) growth; (ii) Employing a critical point theorem due to Ding (Nonlinear Anal. 25(11):1095-1113, 1995), we derive that the system involved has infinitely many weak solutions provided that the potential function is even.
1 Introduction and main results
For \(N\in\mathbb{N}\), let \((\mathbb {R}^{N},\langle\cdot,\cdot\rangle,|\cdot|)\) be the N-dimensional Euclidean space. For fixed \(l,k\in\mathbb{N}\), set \(B:=\{1,2,\ldots, l\} \) and \(C:=\{1,2,\ldots, k\}\). If \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}\) is a smooth function, let ∇f stand for the gradient operator. For a smooth function \(f:\mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow \mathbb {R}\), denote by \(\nabla_{x_{1}}f\) and \(\nabla_{x_{2}}f\) the gradient operator with respect to the first component and the second component, respectively. For a mapping \(f:\mathbb {R}_{+}\rightarrow \mathbb {R}\), \(f(t^{+})\) and \(f(t^{-})\) mean the right-hand side limit and the left-land side limit at t, respectively. For functions \(f:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) and \(g:\mathbb {R}_{+}\rightarrow \mathbb {R}^{n}\), let \(\triangle(f(g(t)))=f(g(t^{+}))-f(g(t^{-}))\).
In this paper, we consider a nonlinear system with impulsive effects on \(\mathbb{H}^{N}:=\mathbb {R}^{N}\times \mathbb {R}^{N}\) for any \(p,q>1\), \(\lambda >0\), \(j\in B\), and \(m\in C\),
with the initial condition \((\dot{u}_{1}(0),\dot{u}_{2}(0))=(u_{1}(0),u_{2}(0))\in\mathbb{H}^{N}\) and the terminal condition \((\dot{u}_{1}(T),\dot{u}_{2}(T))=(0, 0)\in\mathbb{H}^{N}\), where \(\Phi_{\mu}(z):=|z|^{\mu-2}z\) for any \(\mu>1\) and \(z\in \mathbb {R}^{N}\); \(F: \mathbb {R}_{+}\times\mathbb{H}^{N}\to{\mathbb {R}}\); \((t_{j})_{j\in B}\) and \((s_{m})_{m\in C}\) are impulsive times with \(0=t_{0}< t_{1}< t_{2}<\cdots<t_{l}<t_{l+1}=T\), \(0=s_{0}< s_{1}< s_{2}<\cdots<s_{k}<s_{k+1}=T\), and for \(j\in B\) and \(m\in C\), \(I_{j}:{\mathbb {R}}^{N}\to{\mathbb {R}}\) and \(K_{m}:{\mathbb {R}}^{N}\to{\mathbb {R}}\) are continuously differentiable.
For the nonlinear term \(F: [0,T]\times\mathbb{H}^{N}\to{\mathbb {R}}\), we assume that
-
(A1)
For fixed \(t\in[0,T]\) and \(x\in\mathbb{H}^{N}\), \(F(\cdot, x) \) is measurable and \(F(t, \cdot) \) is continuously differentiable;
-
(A2)
There exist \(a_{1},a_{2}\in C({\mathbb {R}}_{+};{\mathbb {R}}_{+})\) and \(b\in L^{1}([0,T];{\mathbb {R}}_{+})\) such that
$$\begin{aligned}& \bigl\vert F(t, x_{1},x_{2})\bigr\vert \leq\bigl[ a_{1}\bigl(\vert x_{1}\vert \bigr)+a_{2}\bigl( \vert x_{2}\vert \bigr)\bigr]b(t), \\& \bigl\vert \nabla F(t,x_{1},x_{2})\bigr\vert \leq \bigl[a_{1}\bigl(\vert x_{1}\vert \bigr)+a_{2} \bigl(\vert x_{2}\vert \bigr)\bigr]b(t), \\& \bigl\vert I_{j}(x_{1})\bigr\vert \leq a_{1} \bigl(\vert x_{1}\vert \bigr), \qquad \bigl\vert \nabla I_{j}(x_{1})\bigr\vert \leq a_{1}\bigl(\vert x_{1}\vert \bigr), \quad j\in B, \\& \bigl\vert K_{m}(x_{2})\bigr\vert \leq a_{2} \bigl(\vert x_{2}\vert \bigr), \qquad\bigl\vert \nabla K_{m}(x_{2})\bigr\vert \leq a_{2}\bigl(\vert x_{2}\vert \bigr), \quad m\in C \end{aligned}$$for all \((x_{1},x_{2})\in{\mathbb{R}}^{N}\times \mathbb {R}^{N}\) and a.e. \(t\in[0, T]\).
For \(N=1\), \(p=q=2\), \(F(t,x_{1},x_{2})=F(t,x_{1})\), and \(I_{j}\equiv0\) (\(j\in B\)), system (1.1) reduces to the following second order impulsive differential equation:
Recently, Chen and Sun [2] investigated the following second order impulsive differential equation:
where \(f\in C(\mathbb {R}_{+}\times\mathbb {R};\mathbb {R})\), \(g, I_{j}\in C(\mathbb {R};\mathbb {R})\). In [2], the authors not only established the variational structure of equation (1.3) but also obtained that (1.3) enjoys three solutions by using an abstract critical point theorem taken from [3]. More precisely, they obtained the following theorem.
Theoremm A
[2], Theorem 3.1
Suppose that
-
(H1)
\(g(u), I_{j}(u)\) are nondecreasing, and \(g(u)u\ge0\), \(I_{j}(u)u\ge0\) for any \(u\in\mathbb {R}\);
-
(H2)
There exist \(a>0,l\in(0,2)\), \(b\in L^{1}(\mathbb {R}_{+};\mathbb {R}_{+})\), and \(c\in L^{2}(\mathbb {R}_{+};\mathbb {R}_{+})\) such that
$$F(t,u)\le b(t) \bigl(a+|u|^{l}\bigr), \qquad f(t,u)\le c(t)|u|^{l-1}, $$for a.e. \(t\ge0\) and \(u\in\mathbb {R}\), where \(F(t,u):=\int_{0}^{u} f(t,s)\,ds\);
-
(H3)
There exist \(d,m,M>0\) such that
$$\frac{d^{2}}{M^{2}}< m^{2}+2\sum_{j=1}^{l} \int_{0}^{me^{-t_{j}}}I_{j}(s)\,ds+2 \int_{0}^{m} g(s)\,ds; $$ -
(H4)
$$\frac{M^{2}\int_{0}^{+\infty}\max_{|\xi|\le d}F(t,\xi)\,dt}{d^{2}}< \frac{\int _{0}^{+\infty}F(t,me^{-t})\,dt}{m^{2}+2\sum_{j=1}^{l}\int _{0}^{me^{-t_{j}}}I_{j}(s)\,ds+2\int_{0}^{m} g(s)\,ds}. $$
Then, for each
(1.3) has at least three classical solutions.
Also, Dai and Zhang [4] showed by using the least action principle that (1.3) has at least one solution if the potential function has subquadratic growth and, by taking advantage of the fountain theorem due to [5], that (1.3) has infinitely many solutions if the potential function is even.
To be precise, they obtained the following theorems.
Theoremm B
[4], Theorem 3.1
Suppose that
-
(S1)
\((I_{j})_{j\in B}\) and g satisfy \(\int_{0}^{u}I_{j}(s)\,ds\ge0\) and \(\int_{0}^{u} g(s)\,ds\ge0\), \(u\in\mathbb {R}\), respectively;
-
(S2)
There exist \(a>0\), \(\alpha\in(1,2)\), and \(b\in L^{1}(\mathbb {R}_{+};\mathbb {R}_{+})\) such that
$$F(t,u)\le b(t) \bigl(a+\vert u\vert ^{\alpha}\bigr) $$for a.e. \(t\ge0\) and all \(u\in\mathbb {R}\).
Then, for \(\lambda>0\), (1.3) has at least one classical solution.
Theoremm C
[4], Theorem 3.2
Besides (S1) above, for a.e. \(t\ge0\) and all \(u\in\mathbb {R}\), assume that
-
(S3)
There exist \(\alpha\in(1,2)\) and \(d\in L^{\frac{2}{2-\alpha }}(\mathbb {R}_{+};\mathbb {R}_{+})\) such that
$$F(t,u)\ge d(t)|u|^{\alpha}; $$ -
(S4)
There exist \(\gamma\in(0,1)\) and \(h_{1},h_{2}\in L^{1}(\mathbb {R}_{+};\mathbb {R}_{+})\) such that
$$f(t,u)\le h_{1}(t)|u|^{\gamma}+h_{2}(t); $$ -
(S5)
There exist \(\gamma_{j}>\alpha-1\), \(\theta>\alpha-1\), and \(q_{j} ,q>0, j\in B\), such that
$$I_{j}(u)\le q_{j}|u|^{\gamma_{j}},\qquad g(u)\le q|u|^{\theta}; $$ -
(S6)
\(f(t, u)\), \(I_{j}(u)\), and \(g(u)\) are odd about u.
Then, for any \(\lambda>0\), (1.3) has infinitely many solutions.
Recently, by applying the least action principle and saddle point theorem, [6–8] investigated the existence of periodic solutions for the following dynamical systems:
and
respectively. Subsequently, by variational approach, Yang and Chen [9, 10] discussed the existence and multiplicity of periodic solutions for the following two classes of nonlinear \((q,p)\)-Laplacian dynamical systems with impulsive effects:
and
respectively, where \(p,q,\lambda,\eta>1\) and \(\rho_{1},\rho_{2},\gamma_{1},\gamma_{2}\in C([0,T];\mathbb {R}_{+})\).
Motivated by [2, 4, 6–10], in this paper, we are interested in the existence and multiplicity of a nontrivial weak solution for system (1.1) by using the least action principle and a critical point theorem due to Ding [1]. To be precise, we obtain the following results.
Theorem 1.1
Suppose that
-
(HIK1)
For \(x_{1},x_{2}\in\mathbb {R}^{N}\),
$$\begin{aligned} \sum_{j=1}^{l} I_{j}(x_{1}) \ge0, \qquad \sum_{m=1}^{k} K_{m}(x_{2})\ge0; \end{aligned}$$ -
(HF1)
There exist \(\alpha_{1}\in[0,q)\), \(\alpha_{2}\in[0,p)\), \(a_{1}>0\), and \(d_{1}\in L^{1}([0,T];\mathbb {R}_{+})\) such that
$$\begin{aligned} F(t,x_{1},x_{2})\le d_{1}(t) \bigl(a_{1}+|x_{1}|^{\alpha_{1}}+|x_{2}|^{\alpha _{2}} \bigr), \quad\forall(x_{1},x_{2})\in\mathbb {R}^{N} \times\mathbb {R}^{N}. \end{aligned}$$
Then, for each \(\lambda>0\), system (1.1) has at least one weak solution in \(X_{q}\times X_{p}\), where, for \(s>1\),
Remark 1.1
There exist examples satisfying Theorem 1.1. For example, let \(q=4\), \(p=3\),
where \(c_{1},c_{2},\xi_{1},\xi_{2}>0\), and for all \(t\in[0,T]\),
or
Theorem 1.2
In addition to (HIK1), we assume that
-
(HF2)
There exist \(a_{2}>0\) and \(d_{2}\in L^{1}([0,T];\mathbb {R}_{+})\) such that
$$\begin{aligned} F(t,x_{1},x_{2})\le d_{2}(t) \bigl(a_{2}+|x_{1}|^{q}+|x_{2}|^{p} \bigr),\quad \forall (x_{1},x_{2})\in\mathbb {R}^{N} \times\mathbb {R}^{N}. \end{aligned}$$
Then, for each \(0<\lambda<\min\{\frac{1}{q(D_{0}(q))^{q}},\frac {1}{p(D_{0}(p))^{p}}\}\), (1.1) has at least one weak solution in \(X_{q}\times X_{p}\), where
Remark 1.2
There exist examples satisfying Theorem 1.2. For example, let \(q=4\), \(p=3\), and \(I_{j}\), \(K_{m}\) defined by (1.8). For all \(t\in[0,T]\), let
or
Theorem 1.3
Along with (HIK1) and (HF2), for \(x_{1},x_{2}\in\mathbb {R}^{N}\), \(j\in B, m\in C\), and \(t\in[0,T]\), we suppose that
-
(HIK2)
There exist \(\nu_{1}\ge q\), \(\nu_{2}\ge q\), and \(\delta_{0}>0\) such that
$$I_{j}(x_{1})\le d_{3}|x_{1}|^{\nu_{1}}, \qquad K_{m}(x_{2})\le d_{4}|x_{2}|^{\nu_{2}}, \quad |x|\le\delta_{0}; $$ -
(HIK3)
\(I_{j}(x_{1})=I_{j}(-x_{1})\), \(K_{m}(x_{2})=K_{m}(-x_{2})\), \(I_{j}(0)=0\), \(K_{m}(0)=0\);
-
(HF3)
There exist \(\mu_{1}\in(1,q)\), \(\mu_{2}\in(1,p)\), \(d_{5}>0\), and \(\delta_{1}>0\) such that
$$\begin{aligned} F(t,x_{1},x_{2})\ge d_{5}\bigl(|x_{1}|^{\mu_{1}}+|x_{2}|^{\mu_{2}} \bigr), \quad |x_{1}|\le \delta_{1},|x_{2}|\le \delta_{1} ; \end{aligned}$$ -
(HF5)
\(F(t,x_{1},x_{2})=F(t,-x_{1},-x_{2}), F(t,0,0)\equiv0 \).
Then, for each \(0<\lambda<\min\{\frac{1}{q(D_{0}(q))^{q}}, \frac {1}{p(D_{0}(p))^{p}}\}\), (1.1) has infinitely many weak solutions in \(X_{q}\times X_{p}\).
Remark 1.3
There exist examples satisfying Theorem 1.3. For example, let \(q=4\), \(p=3\), and
where \(c_{3},c_{4}>0\). For all \(t\in[0,T]\), let
If we take \(\nu_{1}=4.5\), \(\nu_{2}=3.5\), \(\mu_{1}=3.5\), and \(\mu_{2}=2.5\), it is easy to see that the example satisfies Theorem 1.3.
2 Variational structure and some preliminaries
For \(u\in X_{s}\) with \(s=q,p\), define
Set
Set \(X:=X_{q}\times X_{p}\) and define the norm \(\|(u_{1},u_{2})\|_{X}=\|u_{1}\| _{X_{q}}+\|u_{2}\|_{X_{p}}\). Obviously, X is a reflexive Banach space. Let
\(X_{s}\) embeds into \(\mathcal{C}\) continuously and, according to [11], Lemma 2.4,
Lemma 2.1
[12], Proposition 1.2
If \(u_{k}\) converges to u weakly, then \(u_{k}\) uniformly converges to u on \([0,T]\).
If \(u\in X_{s}\), then u is absolutely continuous, whereas u̇ need not be continuous. Hence, it is possible that \(\Delta\Phi_{s}(\dot{u}(t))=\Phi_{s}(\dot{u}(t^{+}))-\Phi_{s}(\dot{u}(t^{-}))\neq0\), which leads to impulse effects.
Following the idea [13], multiplying by \(v_{1}\in X_{q}\) on both sides of the first equation in (1.1) and integrating from 0 to T yields that
Since \(v_{1}\) is continuous, \(v_{1}(t_{j}^{-})=v_{1}(t_{j}^{+})=v_{1}(t_{j})\). Combining \(\dot{u}_{1}(T)=0\) with \(\dot{u}_{1}(0)=u_{1}(0)\) implies that
which, together with (2.2), further leads to
Analogously, for any \(v_{2}\in X_{p}\),
With the two equalities above in hand, we present the notion of weak solutions for (1.1).
Definition 2.1
For any \(v=(v_{1},v_{2})\in X_{q}\times X_{p}\), if
and
then \(u=(u_{1},u_{2})\in X_{q}\times X_{p}\) is called a weak solution of (1.1).
For \(u=(u_{1},u_{2})\in X_{q}\times X_{p}\), define the functional \(\varphi: X\to\mathbb {R}\) by
where
By virtue of (A1) and (A2), by following the argument of [12], Theorem 1.4, one has \(\phi\in C^{1}(X_{q}\times X_{p},{\mathbb{R}})\). Thanks to continuous differentiability of \((I_{j})_{j\in B}\) and \((K_{m})_{m\in C}\), we have \(\psi\in C^{1}(X_{q}\times X_{p},{\mathbb{R}})\). As a consequence, \(\varphi\in C^{1}(X,{\mathbb{R}})\) and, for all \((v_{1},v_{2})\in X_{q}\times X_{p}\),
Definition 2.1 shows that the critical point of φ is the weak solution of system (1.1).
The following lemma plays a crucial role in achieving the critical point of φ.
Lemma 2.2
[14]
Assume that \(\varphi\in C^{1}(E,\mathbb {R})\) is bounded from below (above) and satisfies the (PS) condition. Then
is a critical value of φ.
Lemma 2.3
[1]
Let E be an infinite dimensional Banach space, and let \(\varphi\in C^{1}(E,{\mathbb {R}})\) with \(\varphi(0)=0\) be even and satisfy (PS). If \(E=E_{1}\oplus E_{2}\), where \(E_{1}\) is finite dimensional, and φ satisfies that
- (\(\varphi_{1}\)):
-
φ is bounded from above on \(E_{2}\);
- (\(\varphi_{2}\)):
-
for each finite dimensional subspace \(\tilde{E}\subset E\), there are positive constants \(\rho=\rho(\tilde{E})\) and \(\sigma=\sigma(\tilde{E})\) such that \(\varphi\ge0\) on \(B_{\rho}\cap \tilde{E}\) and \(\varphi|_{\partial B_{\rho}\cap\tilde{E}}\ge\sigma\), where \(B_{\rho}=\{x\in E;\|x\|\le\rho\}\),
then φ possesses infinitely many nontrivial critical points.
3 Proofs of theorems
Proof of Theorem 1.1
It follows from (HIK1), (HF1), and (2.1) that
Owing to \(\alpha_{1}\in[0,q)\) and \(\alpha_{2}\in[0,p)\), we readily obtain that \(\varphi(u)\to+\infty\) as \(\|u\|_{X}\to \infty\), i.e., φ satisfies the coercive condition on X. So φ is bounded below on X.
Hereinafter, we claim that φ satisfies the (PS) condition. If \(\{\varphi(u_{1n},u_{2n})\}\) is bounded and \(\|\varphi'(u_{1n},u_{2n})\|\to 0\) as \(n\to\infty\), then there exists a positive constant \(D_{1}\) such that
Since φ satisfies a coercive condition on X, we infer that \(\|u_{1n}\|_{X_{q}}\) and \(\|u_{2n}\|_{X_{p}}\) is bounded. Next, in light of the reflexive property of \(X_{s}\), there exists a subsequence, still denoted by \(\{u_{n}=(u_{1n},u_{2n})\}\), such that
Thus, Lemma 2.1 gives that
Following the argument in [15–17], we can derive that \(\|u_{n}-u\|_{X}\to0\), where \(u=(u_{1},u_{2})\). Consequently, φ satisfies the (PS) condition. Thus, with the help of Lemma 2.2, we deduce that φ has at least one critical point on X. Hence system (1.1) has at least one solution on X. □
Proof of Theorem 1.2
By (HIK1), (HF2), and (2.1), it follows that
In view of \(\lambda<\min\{\frac{1}{q(D_{0}(q))^{q}},\frac{1}{p(D_{0}(p))^{p}}\} \), one has \(\varphi(u)\to+\infty\) as \(\|u\|_{X}\to\infty\), that is, φ satisfies the coercive condition on X. Hence φ is bounded below on X. By carrying out a similar argument to derive Theorem 1.1, we get that system (1.1) has at least one solution in X. □
Proof of Theorem 1.3
Keep in mind that φ and −φ have the same critical points. Let \(\Theta=-\varphi\). In the sequel, we aim at verifying that all conditions in Lemma 2.3 are fulfilled by Θ. In fact, from (HIK3) and (HF5), we find that Θ is even and \(\Theta(0)=0\). Taking (HIK1), (HF2), and (3.1) into account, we obtain that \(\Theta(u)\to-\infty\) as \(\|u\|_{X}\to\infty\). Hence Θ is bounded above on X so that Θ satisfies (\(\varphi _{1}\)) in Lemma 2.3.
Assume that \(\tilde{X}\subset X\) is finite-dimensional. For any \(u=(u_{1}, u_{2})\in\tilde{X}=\tilde{X}_{q}\times\tilde{X}_{p}\), where \(\tilde{X}_{q}\subset X_{q}\) and \(\tilde{X}_{p}\subset X_{p}\), we deduce that \(\|u_{1}\|_{\mu_{1}}\) is equivalent to \(\|u_{1}\|_{X_{q}}\), and \(\| u_{2}\|_{\mu_{2}}\) is equivalent to \(\|u_{2}\|_{X_{p}}\). Hence there exist constants \(d_{6},d_{7}>0\) such that
Let \(\rho_{0}=\min\{\frac{\min\{\delta_{0},\delta_{1}\}}{D_{0}(q)},\frac{\min\{ \delta_{0},\delta_{1}\}}{D_{0}(p)}\}\). For any \(\rho\in(0,\rho_{0})\), if \(\|u\| _{X}=\rho\), then \(\|u_{1}\|_{\infty}\le D_{0}(q)\|u_{1}\|_{X_{q}}\le D_{0}(q)\rho\le \min\{\delta_{0},\delta_{1}\}\) and \(\|u_{2}\|_{\infty}\le D_{0}(p)\|u_{2}\| _{X_{q}}\le D_{0}(p)\rho\le\min\{\delta_{0},\delta_{1}\}\). Thus it follows from (HIK2), (HF3), (3.2), and Hölder’s inequality that
Observing that \(\mu_{1}\in(1,q)\) and \(\mu_{2}\in(1,p)\), we take sufficiently small \(\rho\in(0,\rho_{0})\) such that \(\Theta(u)\ge0\) on \(B_{\rho}\cap \tilde{X}\) and \(\Theta(u)> 0\) on \(\partial B_{\rho}\cap \tilde{X}\). Therefore, Θ satisfies (\(\varphi_{2}\)) in Lemma 2.3. Then, according to Lemma 2.3, Θ has infinitely many critical points in X so that (1.1) has infinitely many solutions in X. □
References
Ding, YH: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25(11), 1095-1113 (1995)
Chen, H, Sun, J: An application of variational method to second-order impulsive functional differential equation on the half-line. Appl. Math. Comput. 217, 1863-1869 (2010)
Bonanno, G, Marano, SA: On the structure of the critical set of non-differentiable functions with a weak compactness condition. Appl. Anal. 89, 1-10 (2010)
Dai, B, Zhang, D: The existence and multiplicity of solutions for second-order impulsive differential equations on the half-line. Results Math. 63, 135-149 (2013)
Zou, W: Variant fountain theorems and their applications. Manuscr. Math. 104, 343-358 (2001)
Pasca, D: Periodic solutions of a class of nonautonomous second-order differential systems with \((q,p)\)-Laplacian. Bull. Belg. Math. Soc. Simon Stevin 17, 841-850 (2010)
Pasca, D, Tang, CL: Some existence results on periodic solutions of nonautonomous second-order differential systems with \((q,p)\)-Laplacian. Appl. Math. Lett. 23, 246-251 (2010)
Pasca, D, Tang, CL: Some existence results on periodic solutions of ordinary \((q,p)\)-Laplacian systems. J. Appl. Math. Inform. 29(1-2), 39-48 (2011)
Yang, X, Chen, H: Periodic solutions for a nonlinear \((q,p)\)-Laplacian dynamical system with impulsive effects. J. Appl. Math. Comput. 40, 607-625 (2012)
Yang, X, Chen, H: Periodic solutions for autonomous \((q,p)\)-Laplacian system with impulsive effects. J. Appl. Math. 2011, Article ID 378389 (2011)
Zhang, X, Tang, X: Non-constant periodic solutions for second order Hamiltonian system involving the p-Laplacian. Adv. Nonlinear Stud. 13, 945-964 (2013)
Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)
Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10, 680-690 (2009)
Lu, WD: Variational Methods in Differential Equations. Scientific Publishing House in China (2002) (in Chinese)
Xu, B, Tang, CL: Some existence results on periodic solutions of ordinary p-Laplacian systems. J. Math. Anal. Appl. 333, 1228-1236 (2007)
Zhang, X, Tang, X: Periodic solutions for an ordinary p-Laplacian system. Taiwan. J. Math. 15, 1369-1396 (2011)
Yang, X, Chen, H: Existence of periodic solutions for a damped vibration problem with \((q, p)\)-Laplacian. Bull. Belg. Math. Soc. Simon Stevin 21, 51-66 (2014)
Luo, Z, Xiao, J, Xu, Y: Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations. Nonlinear Anal. 75, 2249-2255 (2012)
Zeidler, E: Nonlinear Functional Analysis and Its Applications, Vol. III. Springer, Berlin (1985)
Zhang, X: Subharmonic solutions for a class of second-order impulsive Lagrangian systems with damped term. Bound. Value Probl. 2013, Article ID 218 (2013)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (NO: 61304011) and by the Hunan Provincial Natural Science Foundation of China (NO: 2016JJ3139).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that she has no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Yang, X. Existence and multiplicity of weak solutions for a nonlinear impulsive \((q,p)\)-Laplacian dynamical system. Adv Differ Equ 2017, 128 (2017). https://doi.org/10.1186/s13662-017-1145-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-017-1145-y
MSC
- 34C25
- 58E50
Keywords
- \((q,p)\)-Laplacian
- existence
- multiplicity
- nontrivial solution
- variational methods