# approach to perturb a linear operator

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).

QUESTION.
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