Analog of the eigenvalue problem for degenerate equations (Q1088067)

The author proves the following result: Let f be analytic in the closure of D and \(f\neq 0\) everywhere in D. Consider solutions of \[ (*)\quad div(f\nabla u)+\lambda u=0\quad in\quad D. \] Then, if \(\lambda f<0\) in D and \(f=0\) on \(\partial d\), (*) has no nontrivial solutions which are in \(C'(\bar D)\). This result then leads the author to discuss the values of \(\lambda\) for which (*) has nontrivial solutions. This question in turn leads to interesting relations with special functions and possible extensions of them.