Motivation: Consider the equation, $$-\Delta u = u^p$$ in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,b}(x) = c_n(b/(1+b^2|x-a|^2))^{(n-2)/2}$ upto scaling $b>0$ and translation $a\in \mathbb{R}^n$. It has also been shown that the operator linearized operator is non-degenerate in the sense that any solution of $$-\Delta u = pU_{a,b}^{p-1}u$$ is a linear combination of the derivatives $\partial_a U$ and $\partial_b U$ of $U$ wrt to the parameters $a$ and $b$. The key to proving this result is using separation of variables and then apply Sturm-Liouville Theory.
Question: Now suppose we have equations of the form $$-\operatorname{div}(a(x)\nabla u) = a(x)u^p$$ where $a$ is some weight function (that is non-radial) and say that it also has solutions like the function $U$ up to scaling and translation factor, then what techniques can be used to show that the linearized operator $$-\operatorname{div}(a(x)\nabla v) = pa(x)U^{p-1} v$$ is non-degenerate in the sense that any solution of the above is a linear combination of the derivatives of $U$ wrt to the scaling and translation parameter. Since the weight $a$ is non-radial I don't think a simple separation of variables will work and so probably this problem cannot be resolved by using ODE-based methods. Are there are other well-known tools that could be used here?
