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

Abstract

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), \\ - \partial_t v + (-\Delta)^s v+V(x)v+W(x)u = H_u(t, x, u, v), \\ |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 a wide variety of fields such as theoretical physics, optimal control, chemistry and biology. We obtain ground state solutions of Nehari-Pankov type under mild conditions on the nonlinearity by further developing non-Nehari method with two types of superlinear nonlinearity. If in addition the corresponding functional is even, we also obtain infinitely many geometrically distinct solutions by using some arguments about deformation type and Krasnoselskii genus. Nevertheless, we need to overcome some difficulties: one is that the associated functional is strongly indefinite, the second is due to the absence of strict monotonicity condition, a key ingredient of seeking the ground state solution on suitable manifold, we need some new methods and techniques. The third lies that some delicate analysis are needed for the dropping of classical super-quadratic assumption on the nonlinearity and in verifying the link geometry and showing the boundedness of Cerami sequences.