Let $\Omega = \{(x,y) \in \mathbb{R}^2; \ |x| < \pi/2, \ |y| < \pi/2\}$. The problem $$ \left \{\begin{array}{rcl} - \Delta u - u & = & 0, \ \mbox {in} \ \Omega\\ u(x) & = & g(x), \ \mbox{on} \ \partial \Omega. \end{array} \right.$$ has at most one solution.
Approach idea: As usual, I tried to call $w = v - u$, where $u$ and $v$ are classic and different solutions to the problem. Then we would have: $$ \left \{\begin{array}{rcl} - \Delta w & = & w, \ \mbox {in} \ \Omega\\ w & = & 0, \ \mbox{on} \ \partial \Omega. \end{array} \right.$$ Thus, I tried to use the Poincaré inequality to reach some conclusion. But it was work in vain. Can you help me?