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Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace–Beltrami Equations on Spheres
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- Journal:
- Canadian Mathematical Bulletin / Volume 58 / Issue 4 / 01 December 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 723-729
- Print publication:
- 01 December 2015
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- Article
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We show that a class of semilinear Laplace–Beltrami equations on the unit sphere in
${{\mathbb{R}}^{n}}$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as 
${{\left| s \right|}^{p-1}}s$ for 
$\left| s \right|$ large with 
$1\,<\,p\,<\,\left( n+5 \right)/\left( n-3 \right)$.
