Nonstationary homoclinic solutions for infinite-dimensional fractional reaction-diffusion system with two types of superlinear nonlinearity

This paper is dedicated to studying nonstationary homoclinic solutions with the least energy for a class of fractional reaction-diffusion system \begin{document}$ \begin{eqnarray*} \label{1.1} \left\{\begin{array}{lll} \partial_t u+ (-\Delta)^s u+V(x)u+W(x)v = H_v(t, x, u, v), \\ - v + v+V(x)v+W(x)u H_u(t, |u(t, x)|+|v(t, x)|\rightarrow 0, \ \text{as}\ |t|+|x|\rightarrow \infty, \end{array} \right. \end{eqnarray*} $\end{document} where $ 0<s<1, z (u, v): \mathbb{R}\times \mathbb{R}^N\rightarrow \mathbb{R}^{2} $, which originate from wide variety fields such as theoretical physics, optimal control, chemistry and biology. We obtain ground state Nehari-Pankov type under mild conditions on nonlinearity by further developing non-Nehari method two types superlinear nonlinearity. If in addition corresponding functional even, we also infinitely many geometrically distinct using some arguments about deformation Krasnoselskii genus. Nevertheless, need overcome difficulties: one that associated strongly indefinite, second due absence strict monotonicity condition, key ingredient seeking solution suitable manifold, new methods techniques. The third lies delicate analysis are needed dropping classical super-quadratic assumption verifying link geometry showing boundedness Cerami sequences.