I am stuck with the following problem:
The PDE
$u_{xx}+u_{yy}+\lambda u=0, 0
$u(x,0)=u(x,1)=0; 0\leq x \leq 1$
$u(0,y)=u(1,y)=0; 0\leq y \leq 1$
has
(a)a unique solution u for any $\lambda \in \mathbb R ,$
(b)infinitely many solutions for some $\lambda \in \mathbb R ,$
(c)a solution for countably many values of $\lambda \in \mathbb R ,$
(d)infinitely many solutions for all $\lambda \in \mathbb R .$
I do not know how to progress with it.Could someone point me in the right direction( e.g. a certain theorem or property i have to use?) Thanks in advance for your time.