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Existence of periodic solutions for \(p(t)\)-Laplacian systems. (English) Zbl 1171.34030

Consider the \(p(t)\)-Laplacian system
\[ \begin{aligned} (|u'(t)|^{p(t)-2}u'(t))'+\nabla F(t,u(t))&=0,\\ u(0)-u(T)=u'(0)-u'(T)&=0, \end{aligned}\tag{p} \]
where \(p(t)\in C([0,T],\mathbb R^1)\), \(T>0\), \(F:[0,T]\times\mathbb R^N\to\mathbb R^1\), \(\nabla F(t,x)=\frac{\partial F(t,x)}{\partial x}\) for \(t\in\mathbb R^1\) and \(x\in\mathbb R^N\). In this paper, by using minimax methods in critical point theory, the authors obtained some existence theorems for periodic solutions of the \(p(t)\)-Laplacian system (p). The main results are the following.
Theorem 4.1. Suppose that \(F\) satisfies the following conditions
(A
\(F(t,x)=F(t+T,x)\), \(\nabla F(t,x)\) is continuous for each \(t \in [0,T]\) and \(x\in\mathbb R^N\), \(F(0,0)=0\), \(\int^T_0F(t,x)\,dt\geq 0\) for all \(x\in\mathbb R^N\),
(A
there exist \(\beta>p^+\) and \(r_1>0\) such that \((\nabla F(t,x),x)\geq \beta F(t,x)\) for \(|x|\geq r_1,\) where \((\cdot,\cdot)\) is the usual inner product of \(\mathbb R^N\) and \(p^+=\max_{t\in[0,T]}p(t)\),
(A
there exist \(\mu>p^+\) and \(g\in C([0,T],\mathbb R^1)\) such that \(\limsup_{|x|\to 0}\frac{|F(t,x)|}{|x|^\mu}\leq|g(t)|\),
(A
\(p(t)\in C([0,T],\mathbb R^1)\), \(p^-=\min_{t\in[0,T]}p(t)>1\) and \(p(t)=p(t+T)\) for all \(t\in\mathbb R^1\).
Then system (p) has at least one periodic solution.
Theorem 4.2. Suppose that \(F\) satisfies (A\(_1)\)–(A\(_4\)) and \(F(t,x)=F(t,-x)\) for all \(t\in\mathbb R^1,x\in\mathbb R^N.\) Then system (p) has infinite many periodic solutions.
Theorem 4.3. Suppose that \(F\) satisfies the following conditions
(B
\(F(t,x)=F(t+T,x)\), \(\nabla F(t,x)\) is continuous for each \(t\in [0,T]\) and \(x\in\mathbb R^N,\)
(B
there exist \(\eta\in (0,p^-)\) and \(r_2>0\) such that \((\nabla F(t,x),x)\leq \eta F(t,x)\) for \(|x|\geq r_2,\) where \((\cdot,\cdot)\) is the usual inner product of \(\mathbb R^N,\)
(B
\( \int_0^TF(t,x)\,dt\to\infty\) as \(|x|\to\infty\),
(B
\(p(t)\in C([0,T],\mathbb R^1)\), \(p^-=\min_{t\in[0,T]}p(t)>1\) and \(p(t)\) is \(T\)-periodic.
Then system (p) has at least one periodic solution.
In some case, Theorem 4.3 generalizes the results in [B. Xu and C.-L. Tang, J. Math. Anal. Appl. 333, No. 2, 1228-1236 (2007; Zbl 1154.34331)].

MSC:

34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Citations:

Zbl 1154.34331
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References:

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