# Shooting from singularity to singularity and a semilinear Laplace–Beltrami equation

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## Abstract

For surfaces of revolution we prove the existence of infinitely many rotationally symmetric solutions to a wide class of semilinear Laplace–Beltrami equations. Our results extend those in Castro and Fischer (Can Math Bull 58(4):723–729, 2015) where for the same equations the existence of infinitely many even (symmetric about the equator) rotationally symmetric solutions on spheres was established. Unlike the results in that paper, where shooting from a singularity to an ordinary point was used, here we obtain regular solutions shooting from a singular point to another singular point. Shooting from a singularity to an ordinary point has been extensively used in establishing the existence of radial solutions to semilinear equations in balls, annulii, or \(\mathbb {R}^N\).

## Keywords

Laplace–Beltrami operator Pohozaev identity Semilinear equation Rotationally symmetric solution Superlinear nonlinearity Subcritical nonlinearity## Mathematics Subject Classification

35J25 58J05## Notes

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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