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Almost global existence for exterior Neumann problems of semilinear wave equations in $2$D
November 2013, 12(6): 2361-2380. doi: 10.3934/cpaa.2013.12.2361
Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance
Shiwang Ma 1,
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071
Received July 2011 Revised May 2012 Published May 2013
In this paper, we consider the existence of nontrivial $1$-periodic solutions of the following Hamiltonian systems \begin{eqnarray} -J\dot{z}=H'(t,z), z\in R^{2N}, \end{eqnarray} where $J$ is the standard symplectic matrix of $2N\times 2N$, $H\in C^2 ( [0,1] \times R^{2N}, R)$ is $1$-periodic in its first variable and $H'(t,z)$ denotes the gradient of $H$ with respect to the variable $z$. Furthermore, $H'(t,z)$ is asymptotically linear both at origin and at infinity. Based on the precise computations of the critical groups, Maslov-type index theory and Galerkin approximation procedure, we obtain some existence results for nontrivial $1$-periodic solutions under new classes of conditions. It turns out that our main results improve sharply some known results in the literature.
Keywords: periodic solution, Critical group, resonance., Galerkin approximation, Maslov-type index.
Mathematics Subject Classification: Primary: 34B15, 34C25, 37J4.
Citation: Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361
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Shiwang Ma