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arXiv · 1805.04321

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

Anna Lisa Amadori·Francesca Gladiali
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Abstract

We investigate nodal radial solutions to semilinear problems of type \[\begin{cases}-\Delta u = f(|x|,u) \qquad & \text{ in } \Omega, \newline u= 0 & \text{ on } \partial \Omega, \end{cases} \] 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.

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Anna Lisa Amadori, Francesca Gladiali. 2018-05-11. On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE's. https://arxiv.org/abs/1805.04321

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