Dirichlet problem, uniqueness & counterexample Help me please with this example:
Let's $q(x)$ be a continuous and constant sign function in $[0,1]$;
uniqueness of solution of Dirichlet problem to equation: $u''+qu=0$ depends on sign of $q$.
Prove the theorem of uniqueness in cases then it's true and give counterexample in case then it's not true.
Thanks a lot!

Without boundary conditions like $u(0)=a$, $u(1)=b$ you will have no chance proving uniqueness. So from now on I'll assume you have boundary conditions given. 
If $q\geq 0$ you want have a unique solution in general. A counterexample would be the follwing: I'm looking for solutions to $u''+u=0$, $u(0)=0$, $u(\pi)=0$. It's clear that $u(x)=0$ und $u(x)=sin(x)$ satisfy these conditions.
Assumung $q\leq 0$ the key step in proving uniqueness is the so called maximum principle. It states that the maximum (under certain conditions, i.e. $q\leq 0$, http://aw.twi.tudelft.nl/~sweers/maxpr/maxprinc.pdf) can only be attained on the boundary. Assuming that $\phi$ and $\psi$ are solutions then $\phi-\psi$ is also a solution which can attain it's maximum only on the boundary. But since $\phi=\psi$ on the boundary this implies $\phi=\psi$ everywhere.
A good reference are chapters 1 and 2 in http://books.google.de/books?id=2qMa1UgeeNcC&printsec=frontcover&hl=de#v=onepage&q&f=false
