My question is related to how one normally would perturb a linear operator.

Let $B_1$ denote the open unit ball in $ R^N$ and suppose $\gamma>0$ is such that the operator $$L(\phi):=\Delta \phi(x) + \frac{ \gamma \phi(x)}{|x|^2} = f(x) \quad \mbox{ in } B_1\backslash \{0\} $$ with $ \phi=0$ on $ \partial B_1$ has good solvability properties. So for instance lets suppose there is some $ \sigma>0$ (assume small) such that for all $ f \in Y$ we can find a unique $ \phi \in X$ which satisfies the above and one has an estimate of the form $ \| \phi \|_X \le C \|f\|_Y$ where $ \| \phi \|_X:=\sup_{B_1} |x|^\sigma | \phi(x)|$ and $ \| f \|_Y:= \sup_{B_1} |x|^{\sigma+2} |f(x)|$. (to obtain a result like this one would write everything in terms of spherical harmonics).

Consider $0 \subset D \subset B_{1/100}$ is some open set and for $ \varepsilon>0$ small set $\Omega_\varepsilon:= B_1 \backslash \overline{ \varepsilon D}$. I would like to have a theory that says there is some $ \varepsilon_0>0$ (small) and $C>0$ such that for all $ 0<\varepsilon < \varepsilon_0$ and all (the following function spaces are restricted to functions on $ \Omega_\varepsilon$)
$ f \in Y$ there is some $ \phi \in X$ such that $L(\phi)=f$ in $\Omega_\varepsilon$ with $ \phi=0$ on $ \partial \Omega_\varepsilon$ and $ \| \phi \|_X \le C \|f\|_Y$.

Approaches I know. Suppose $D \approx B_\frac{1}{100}$ and I know how to solve the problem on the annulus $ \{x: \frac{1}{100} <|x|<1 \}$ then I can do a change of variables and solve it on perturbations of the annulus.

I have seen various papers solve problems like the above by replacing the scalar equation with a system and they solve the problem this way (I know I am being vague here). This approach requires various cut offs.

QUESTION. I would prefer an approach where I view the term $ \frac{\gamma \phi}{|x|^2}$ as a compact perturbation of $\Delta$ and apply some Fredholm theory. Then one can assume the result is false and hopefully in the limit get something in the kernel of $L$ on the punctured ball (which contradicts the original assumption on $L$). With this approach there seems to be a lot of cases to consider and I can't tell if this works or not. If it works I thought this is a more direct approach then I have seen and hence I would assume it must not work (otherwise people would have done it...)

Sorry i know the question is not well posed. I am looking more for some comments abouts about what the normal approaches to handle things like this are and whether this Fredholm approach would work or thanks


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