Infinitely many periodic solutions for second order Hamiltonian systems

In this paper, we study the existence of infinitely many periodic solutions for second order Hamiltonian systems $\ddot{u}+\nabla_u V(t,u)=0$, where $V(t, u)$ is either asymptotically quadratic or superquadratic as $|u|\to \infty$.Comment: to appear in JDE(doi:10.1016/j.jde.2011.05.021

A Note on Homoclinic Orbits for Second Order Hamiltonian Systems

In this paper, we study the existence for the homoclinic orbits for the second order Hamiltonian systems. Under suitable conditions on the potential $V$, we apply the direct method of variations and the Fourier analysis to prove the existence of homoclinc orbits

New existence and multiplicity of homoclinic solutions for second order non-autonomous systems

In this paper, we study the second order non-autonomous system \begin{eqnarray*} \ddot{u}(t)+A\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \ \forall t\in\mathbb{R}, \end{eqnarray*} where $A$ is an antisymmetric $N\times N$ constant matrix, $L\in C(\mathbb{R},\mathbb{R}^{N\times N})$ may not be uniformly positive definite for all $t\in\mathbb{R}$, and $W(t,u)$ is allowed to be sign-changing and local superquadratic. Under some simple assumptions on $A$, $L$ and $W$, we establish some existence criteria to guarantee that the above system has at least one homoclinic solution or infinitely many homoclinic solutions by using mountain pass theorem or fountain theorem, respectively. Recent results in the literature are generalized and significantly improved

Homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign nonlinearities

In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byproduct, we obtain the uniqueness of the positive ground state solution for a class of autonomous Hamiltonian systems and the best constant for Sobolev inequality which are of independent interests

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