Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves $$\begin{cases} -\Delta u_0 + \frac{1}{r^2} u_0 = 0, & \text{on $M$} \\ 5u_0 + \partial_ru_0 = h, & \text{on $\partial M$} \end{cases}$$ Let $C = \sup |u_0|$.
Let $u_1$ be the unique function that vanishes at infinity and solves $$\begin{cases} -\Delta u_1 + \frac{1}{r^2} u_1 = u_0|_{\partial M}-C, & \text{on $M$} \\ 5u_1 + \partial_ru_1 = h, & \text{on $\partial M$} \end{cases}$$ where $u_0|_{\partial M}$ is the function $(r, p) \mapsto u_0(1, p)$ for $p \in \partial M$.
Define a sequence of functions $u_n$ in the following way: for $n \geq 2$, $u_n$ is the unique function that vanishes at infinity and solves $$\begin{cases} -\Delta u_n + \frac{1}{r^2} u_n = u_{n-1}|_{\partial M}, & \text{on $M$} \\ 5u_n + \partial_ru_n = h, & \text{on $\partial M$} \end{cases}$$
By the maximum principle, we have that $u_0 \geq u_1 \geq u_2\geq ...$.
Is any of the following true?
- $\lVert u_n \rVert_{L^2(\partial M)}$ uniformly bounded.
- $u_n$ are uniformly bounded from below.
- $u_n$ converges to a function in some space (a weighted sobolv space or a weighted holder space).
References are appreciated.
