Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>1 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $. 

Since we have already known that the harmonic function have the unique continuation property, I guess such result is still true for the semilinear case here. However I do not know how to go on. Can you give me some hints or references?