It is related to the classic theory of PDE.
Suppose that $U \subset \mathbb{R}^m$ is an open, connected and bounded set, $f : U \to \mathbb{R}$ and $g:\partial U\rightarrow \mathbb{R}$ and $c:\overline{U} \to \mathbb{R}$, with $f, g, c$ continuous functions. Let be following problem
$${}\begin{cases} \Delta v +c(x) v= f& in &U\\ \frac{\partial v}{\partial \nu}=g & in &\partial U\\ \end{cases}$$
where $\nu$ is the outer normal exterior. Prove that, under suitable assumptions, this problem can have infinitely many solutions in $C^2(U)\cap C^1(\overline{U})$. In the solution, I must use steps one and two below in order to obtain infinite solutions for the problem above!!
My attempt to solve was based on three steps:
Replace the hypothesis in the problem above for the corresponding Dirichlet problem.
I applied the Schauder Theorem of existence to prove that exists a unique solution.
Now I am struggling to conclude that from steps one and two I will obtain infinite solutions for the problem in this question.
Would you give me some hints? Someone told me that I have to consider the Fredhom Alternative with a fixed point different from identity, but, I couldn't get this.