The existence of periodic solution for infinite dimensional Hamiltonian systems

In this paper, we will consider a kind of infinite dimensional Hamiltonian system(HS), by the method of saddle point reduction, topology degree and the index, we will get the existence of periodic solution for (HS)

Ground state solutions for diffusion system with superlinear nonlinearity

In this paper, we study the following diffusion system \begin{equation*} \begin{cases} \partial_{t}u-\Delta_{x} u +b(t,x)\cdot \nabla_{x} u +V(x)u=g(t,x,v),\\ -\partial_{t}v-\Delta_{x} v -b(t,x)\cdot \nabla_{x} v +V(x)v=f(t,x,u) \end{cases} \end{equation*} where $z=(u,v)\colon\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $b\in C^{1}(\mathbb{R}\times\mathbb{R}^{N}, \mathbb{R}^{N})$ and $V(x)\in C(\mathbb{R}^{N},\mathbb{R})$. Under suitable assumptions on the nonlinearity, we establish the existence of ground state solutions by the generalized Nehari manifold method developed recently by Szulkin and Weth