What are the number of solutions for the given system of partial differential equation? Suppose $u\in C^2(\bar B),B$ is the unit ball in $\mathbb R^2,$ satisfies $$\Delta u=f \:\text{ in  }B$$
$$\alpha u+\frac{\partial u}{\partial n}=g \:\text{ on } \partial B \ , \alpha>0$$ Where $n$ is the unit outward normal to $B.$If a solution exists then
(1) it is unique.
(2) there are exactly two solutions.
(3) there are exactly three solutions.
(4) there are infinitely many solutions. 
Solution:
From the above question, all I understood is $u$ is a continuous function on a unit disk($B$) in $\mathbb R^2$.


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*What are the references for the above type of problems?

*How to initiate the above problem.

*Need hint.


First, notice that if the PDE has at least two distinct solutions $u_1$ and $u_2$, then $\theta u_1 + (1-\theta)u_2$ is also a solution for any real number $\theta$.  Thus, the only possibility is that the PDE either has one or infinitely many solutions (if is has at least one).
I'll give you a hint for determining if there is only one solution or infinitely many solutions:
Hint: Suppose there are two solutions $u_1$ and $u_2$ to the problem.  What PDE does $w = u_1 - u_2$ solve?  Can you compute
$$\int_B w \Delta w\;dx?$$
What does this tell you?
