## 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 \(\mathbb {R}^N\).

This is a preview of subscription content, log in to check access.

## References

- 1.
Azzollini, A.: On a prescribed mean curvature equation in Lorentz–Minkowski space. J. Math. Pures Appl. (9)

**106**(6), 1122–1140 (2016) - 2.
Bartsch, T., Wang, Z.Q.: On the existence of sign changing solutions for semilinear Dirichlet problems. Topol. Methods Nonlinear Anal.

**7**, 115–131 (1996) - 3.
Bonanno, G., Bisci, G.M., Radulescu, V.: Nonlinear elliptic problems on Riemannian manifolds and applications to Emden–Fowler type equations. Manuscr. Math.

**142**(1–2), 157–185 (2013) - 4.
Boscaggin, A., Dambrosio, W., Papini, D.: Multiple positive solutions to elliptic boundary blow-up problems. J. Differ. Equ.

**262**(12), 5990–6017 (2017) - 5.
Castro, A., Cossio, J., Neuberger, J.M.: Sign changing solutions for a superlinear Dirichlet problem. Rocky Mt. J. M.

**27**(4), 1041–1053 (1997) - 6.
Castro, A., Cossio, J., Neuberger, J.M.: On multiple solutions of a nonlinear Dirichlet problem. Nonlinear Anal. TMA

**30**(6), 3657–3662 (1997) - 7.
Castro, A., Fischer, E.M.: Infinitely many rotationally symmetric solutions to a class of semilinear Laplace–Beltrami equations on spheres. Can. Math. Bull.

**58**(4), 723–729 (2015) - 8.
Castro, A., Kurepa, A.: Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball. Proc. Am. Math. Soc.

**101**(1), 57–64 (1987) - 9.
Guillemin, V., Pollak, A.: Differential Topology. American Mathematical Society, New York (2014)

- 10.
Kristaly, A., Papageorgiou, N.S., Varga, C.: Multiple solutions for a class of Neumann elliptic problems on compact Riemannian manifolds with boundary. Can. Math. Bull.

**53**(4), 674–683 (2010) - 11.
Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence (1986)

## Author information

### Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

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

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by a grant from the Simons Foundation (# 245966 to Alfonso Castro). I. Ventura was partially supported by the NSF under Grant DMS-0946431.

## Appendix A: Computation of the Laplace–Beltrami operator on surfaces of revolution in geodesic metric

### Appendix A: Computation of the Laplace–Beltrami operator on surfaces of revolution in geodesic metric

In order to prove that the solutions to (1.5) are the rotationally symmetric solutions to (1.3) first we compute \(\partial ^2_{x_N}u\) for functions that are constant along normal directions to *M*. More precisely, let \(\hat{s} \in (0,a)\). Since rotationally symmetric functions satisfy \(u(\theta , x_N) = u(0, |\theta |, z)\) we simply write \(u(0,x_{N-1}, x_N) = u(x_{N-1}, x_N)\). Let \((0, \ldots , 0, G(t), z(t)):= (G(t), z(t))\) denote a local parametrization of

around \((0, \ldots , 0, G(\hat{s}) , z(\hat{s}))\) such that \((G'(s))^2 + (z'(s))^2 = 1\). By the \(\epsilon \)-neighborhood theorem, see [9], there exists \(\delta >0\) and differentiable functions \(s:(-\delta , \delta ) \rightarrow \mathbb {R}\) and \(t:(-\delta , \delta ) \rightarrow \mathbb {R}\) such that

for all \( h \in (-\delta , \delta )\), \(s(0) = \hat{s}\) and \( t(0) = 0\). Differentiating in (A.1) with respect to *h* we have

Taking \(h=0\) and using that \((G'(s))^2 + (z'(s))^2 = 1\) yield \(t'(0) = G'(\hat{s})\) and \(s'(0) =z'(\hat{s})\). Differentiating again gives

which in turn implies

Assuming now that *u* is constant along normal lines in an \(\epsilon \)-neighborhood of *M* and defining \(v(s) = u((G(s),z(s)) + t\eta (s))\) with \(\eta (s) = (-z' (s), G'(s))\) we have

Thus, by L’Hopital rule,

Now we proceed to the computation of \(\frac{\partial ^2u}{\partial x_{N-1}^2}\). Using again the \(\epsilon \)-neighborhood theorem, for *h* small enough there exist differentiable functions *s*(*h*) and *t*(*h*) such that

Differentiating with respect to *h*,

Also, \(t(0) = 0\) and \(s(0) = \hat{s}\). Hence \(1 = G'(\hat{s}) s'(0) - t'(0) z'(\hat{s})\) and \( 0= z'(\hat{s}) s'(0) + t'(0) G'(\hat{s})\). Thus

Differentiating in (A.8) with respect to *h*, replacing *h* by 0 and using (A.9) we have

Using that \(z'(s)z''(s) = - G'(s) G''(s)\) for all *s* we have

Since *u* is constant along normal lines in an \(\epsilon \)-neighborhood of *M*, defining \(v(s) = u(0,G(s),z(s)) + t\eta (s))\) with \(\eta (s) = ( 0,-z' (s), G'(s))\) we have

Thus, by L’Hopital rule,

From (A.9), (A.6), and (A.12),

where we have used that the Laplacian operator \(\Delta \) in \(\mathbb {R}^{N-1}\) for radial functions is given by \(\partial ^2_r+ ((N-2)/r)\partial _r\). This proves that rotational symmetric solutions to (1.3) are given by solutions to (1.5) subject to (1.6).

## Rights and permissions

## About this article

### Cite this article

Castro, A., Ventura, I. Shooting from singularity to singularity and a semilinear Laplace–Beltrami equation.
*Boll Unione Mat Ital* **12, **557–567 (2019). https://doi.org/10.1007/s40574-019-00191-y

Received:

Accepted:

Published:

Issue Date:

### Keywords

- Laplace–Beltrami operator
- Pohozaev identity
- Semilinear equation
- Rotationally symmetric solution
- Superlinear nonlinearity
- Subcritical nonlinearity

### Mathematics Subject Classification

- 35J25
- 58J05