## 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

Zbl 1154.34331
Full Text:

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