Uniqueness of $ \Delta u= \int _\Omega u(y)dy $ , $ x \in \Omega $

Question

How can i show uniqueness of

$$ \Delta u= \int _\Omega u(y)dy \phantom{2} ,\phantom{2} x \in \Omega $$ $$ u=0 \phantom{2}, \phantom{2} x \in \partial\Omega $$

I suppose that there are two solutions $u_1$ and $u_2$ such that $w=u_1 - u_2$. Applying Green first identity , i cannot get $w=0$

Answer 1

Multiply the equation by $u$ and integrate over $\Omega$. Integrating by parts on the left hand side we have $$ -\int_\Omega |\nabla u|^2 = \left(\int_\Omega u\right)^2. $$ Note that the left hand side is non-positive and the right hand side is non-negative. Thus it must be the case that $u \equiv 0$.