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A research problem led me to consider the following modification of Bessel's equation: $$ - \xi''(r) - \frac{N - 1}{r} \xi'(r) - \left(f(r) - \frac{\theta}{r^2} \right)\xi(r) = 0 \quad \text{in} \quad [0, 1] $$ with boundary conditions $$ \xi'(0) = 0, \ \xi(1) = 1. $$

Of course some hypotheses on $f$ are required, but I can maybe adapt the problem to them.

Is there any theory regarding such equations with the term $f(r)$ in place of the $1$ that appears in the classical Bessel equation? Any reference? In fact, it would suffice to have some information on $\xi'(1)$ or $\xi''(1)$.

Thanks in advance.

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    $\begingroup$ if $f(r)=r^p$ the solution is still a Bessel function, with argument $r^{1+p/2}$. $\endgroup$
    – Carlo Beenakker
    Commented Jul 18, 2022 at 19:42
  • $\begingroup$ @CarloBeenakker thank you. Unfortunately this is too restrictive: $f(r) = g'(u(r))$, where $u$ is the radial solution of $- \Delta u = g(u)$ in the unit ball (with Dirichlet condition). Maybe some specific nonlinearity $g$ would give some good information? $\endgroup$
    – Danilo Gregorin Afonso
    Commented Jul 19, 2022 at 9:46

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