Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems. (English) Zbl 1189.35091
Summary: This paper is concerned with the following periodic Hamiltonian elliptic system
\[ \begin{cases} -\Delta\varphi+ V(x)\varphi= G_\psi(x,\varphi,\psi) &\text{in }\mathbb R^N,\\ -\Delta\psi+ V(x)\psi= G_\varphi(x,\varphi,\psi) &\text{in }\mathbb R^N,\\ \varphi(x)\to 0\;\text{ and }\;\psi(x)\to 0 &\text{as }|x|\to\infty. \end{cases} \]
Assuming the potential \(V\) is periodic and 0 lies in a gap of \(\sigma(-\Delta+V)\), \(G(x,\eta)\), is periodic in \(x\) and asymptotically quadratic in \(\eta=(\varphi,\psi)\), existence and multiplicity of solutions are obtained via variational approach.
\[ \begin{cases} -\Delta\varphi+ V(x)\varphi= G_\psi(x,\varphi,\psi) &\text{in }\mathbb R^N,\\ -\Delta\psi+ V(x)\psi= G_\varphi(x,\varphi,\psi) &\text{in }\mathbb R^N,\\ \varphi(x)\to 0\;\text{ and }\;\psi(x)\to 0 &\text{as }|x|\to\infty. \end{cases} \]
Assuming the potential \(V\) is periodic and 0 lies in a gap of \(\sigma(-\Delta+V)\), \(G(x,\eta)\), is periodic in \(x\) and asymptotically quadratic in \(\eta=(\varphi,\psi)\), existence and multiplicity of solutions are obtained via variational approach.
MSC:
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35J50 | Variational methods for elliptic systems |
| 35J47 | Second-order elliptic systems |
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