Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>2 $ 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?
Your functions $u_j$ are solutions of a semi-linear elliptic equation and, for $p>2$ the function $t\mapsto\vert t\vert^{p-1}t=\phi(t)$ is $C^2$; as a consequence, each $u_j$ is $C^\alpha$ with some positive $\alpha$. We have also with $w=u_2-u_1$ $$ ∆w= \phi(u_1)-\phi(u_1+w) $$ so that, since $u_1, w$ are continuous and $\phi'$ is locally bounded $$ \vert ∆ w\vert\le C\vert w\vert, \quad\text{$w=0$ on $A$.} $$ This is a standard case of unique continuation from an open subset. It would be even enough to assume that $u_2-u_1$ has a flat point inside $B_1$, i.e. is such that $$ \int_{B(x_0,r)}\vert w(x)\vert dx =O(r^N)\text{ for any $N$, when $r$ goes to 0}. $$
