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This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term$$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.

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This paper is concerned with a class of fractional Schr\”{o}dinger equation with Hardy potential \begin{equation}\nonumber (-\Delta)^{s}u+V(x)u-\frac{\kappa}{|x|^{2s}}u=f(x,u),~~x\in \mathbb{R}^{N}, \end{equation} where $s\in(0,1)$ and $\kappa\geq0$ is a parameter. Under some suitable conditions on the potential $V$ and the nonlinearity $f$, we prove the existence of ground state solutions when the parameter $\kappa$ lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about $\kappa$. Finally, we are able to explore the asymptotic behaviors of ground state solutions as $\kappa$ tends to $0$.