Some nonexistence results of stationary solutions for the Gray–Scott model

We study some stationary problems for the Gray–Scott model. The existence of stationary solutions has been established by Hale et al. [J.K. Hale, L.A. Peletier, W.C. Troy, Exact homoclinic and heteroclinic solutions of the Gray–Scott model for autocatalysis, SIAM J. Appl. Math. 61 (2000) 102–130]. In this paper we will show some sufficient conditions about the nonexistence of nontrivial stationary solutions. To prove the result, we need some a priori estimates derived by using the strong maximum principle.