A priori estimates of stationary solutions of an activator-inhibitor   system
  We consider positive solutions of the stationary Gierer-Meinhardt system. Under suitable conditions on the exponents $p,q,r$ and $s$, different types of a priori estimates are obtained, existence and non-existence results of nontrivial solutions are derived, for both positive and zero source production cases. 
Huiqiang Jiang and Wei-Ming Ni