Abstract This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term −Δu+b→(x)⋅∇u+V(x)u=Hv(x,u,v)inRN,−Δv−b→(x)⋅∇v+V(x)v=Hu(x,u,v)inRN. $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$ 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.

中文翻译：

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带梯度项的哈密顿椭圆系统的基态和多重解

摘要 本文涉及以下非线性哈密顿椭圆系统，其梯度项为−Δu+b→(x)⋅∇u+V(x)u=Hv(x,u,v)inRN,−Δv−b→(x )⋅∇v+V(x)v=Hu(x,u,v)inRN。$$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+ V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\ Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R }^{N}.\\ \end{array} \right。\end{array}$$ 与现有的一些问题相比，本文最有趣的特点是我们假设非线性满足局部超二次条件，比通常的全局超二次条件弱。这种情况允许非线性在某些域上是超二次的，而在其他域上是渐近二次的。此外，通过使用变分方法，我们在局部超二次条件下获得了基态解和无限多个几何不同解的新存在结果。由于我们没有关于非线性的更多全局信息，因此在证明中我们应用了扰动方法和一些特殊技术。