# Eigenvalues of elliptic operator analytic with respect to a parameter

I am interested when one can say the eigenvalues of an elliptic operator are real analytic with respect to a parameter. In particular I have seen many people say the first eigenvalue is analytic but for the higher eigenvalues there can be problems (when the eigenvalues cross).

Let me state the setting I am in now. Consider $$-\Delta w_p(r) = g(r) w_p(r) + (w_p(r))^p$$ in $B_1$ (the unit ball in $R^N$) with $w_p(1)=0$. Here $p>1$ is a parameter and we assume $g(r)$ is analytic in $r$. We assume for $p$ in some given interval we have a unique radial solution which I am writing as $w_p$ and we assume this solution is radial nondegenerate. I am interested when I can say the solution truly is nondegenerate (ie. want the kernel of the linearized operator to be trivial; it is trivial over the space of radial functions by assumption).

So what I would like is that the eigenvalues $\mu_k(p)$ are analytic with respect to $p$. Lets assume we are looking at the second eigenvalue. Since we have symmetry i assume the second eigenvalue will have large multiplicity (maybe) and hence we shouldn't have to worry about any crossings of eigenvalues. I realize this question isn't well posed; but any comments would be greatly appreciated.

I am aware of the paper ''Real analyticity and non-degeneracy'' by Norman Dancer in Mathematische Annalen.

• This looks like it might be a question where one starts with T. Kato, Perturbation Theory of Linear Operators – Neal May 15 '17 at 19:30
• @ChristianRemling Touche; I have edited my comment :) – Neal May 15 '17 at 19:35
• @Neal Thanks for the reference I will take a look. – Math604 May 15 '17 at 19:51