If $\int\limits_V f \; \mathrm dV = 0$ can we say that $f=0$ everywhere? Or what conditions are there on concluding this.
In particular I want to solve the PDE $\nabla^2 f=f^3$ on the region $$D=\{(x,y)\in\mathbb{R}|x^2+y^2<1\}$$ with $f=0$ on the boundary.