The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems II

This paper is a continuation of Zhong et al. (2014), we go on discussing some properties of the global attractor for some symmetric dynamical system with a Lyapunov function F in a Banach space. The difference between this paper and Zhong et al. (2014) is that the origin is not a local minimum point but a saddle point of F. Under some suitable assumptions, we establish an abstract result about the existence of the multiple equilibrium points in the global attractor by estimating the lower bound of Z2-index of the global attractor. As an application of this abstract result, we consider the existence of multiple stationary solutions for some semilinear reaction-diffusion equation when the origin is an unstable equilibrium point.