On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's

Abstract

We investigate nodal radial solutions to semilinear problems of type \[\begin{cases}-Δu = f(|x|,u) \qquad & \text{ in } Ω, \newline u= 0 & \text{ on } \partial Ω, \end{cases} \] where $\Omega$ is a bounded radially symmetric domain of ${\mathbb{R}}^N$ ($N\ge 2$) and $f$ is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse index in a forthcoming work.

Cited by in corpus (1)

• The Hénon problem with large exponent in the disc