What are the conditions[General Criteria] for the existence or non existence of the solutions to a PDE[Elliptic type] subject to given boundary conditions?
A specific Example:
Let's consider the reduced wave equation: $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=-k^2u$$
In the simplest case if u vanishes on the square:$x=0,x=a,y=0,y=a$ the solution is: $$u=\sin\frac{m\pi x}{a}\sin\frac{n\pi\ y}{a}$$
where, $k^2=(m^2+n^2)\pi^2/a^2$.
Can we use this result to solve: $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=f(u)$$
One may to expand f(u) in terms of the eigenfunctions of the operator:
$$\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}$$ How should one go about the job if the boundary conditions on the same square are changed to a different continuous and differentiable functions of the type:
Example: For x=0 $u=f_1(x,y)$
For x=a $u=f_2(x,y)$
For y=0 $u=f_3(x,y)$
For x=a $u=f_4(x,y)$