Homoclinic orbits for an unbounded superquadratic

We consider the following nonperiodic diffusion systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{\begin{array}{ll} \partial_{t}u-\triangle_{x}u+b(t,x)\nabla_{x}u+V(x)u=G_{v} (t,x,u,v), \\ -\partial_{t}v-\triangle_{x}v-b(t,x)\nabla_{x}v+V(x)v=G_{u} (t,x,u,v), \end{array}\right. {\forall}(t,x)\in\mathbb{R} \times\mathbb{R}^{N}, $$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b\in C(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R}^{N}), G\in C^{1} (\mathbb{R}\times\mathbb{R}^{N}\times\mathbb{R}^{2m},\mathbb{R})}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${z:=(u,v): \mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{m}\times\mathbb{R}^{m}}$$\end{document}. Suppose that the potential V is positive constant and G(t, x, z) is superquadratic in z as |z| → ∞. By applying a generalized linking theorem for strongly indefinite functionals, we obtain homoclinic solutions z satisfying z(t, x) → 0 as |(t, x)| → ∞.