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Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace-Beltrami Equations on Spheres

  Published:2015-08-04
 Printed: Dec 2015
  • Alfonso Castro,
    Department of Mathematics , Harvey Mudd College , Claremont, CA 91711 , USA
  • Emily Fischer,
    Department of Mathematics , Harvey Mudd College , Claremont, CA 91711, USA
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Abstract

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 $|s|^{p-1}s$ for $|s|$ large with $1 \lt p \lt (n+5)/(n-3)$.
Keywords: Laplace-Beltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, sub-super critical nonlinearity Laplace-Beltrami operator, semilinear equation, rotational solution, superlinear nonlinearity, sub-super critical nonlinearity
MSC Classifications: 58J05, 35A24 show english descriptions Elliptic equations on manifolds, general theory [See also 35-XX]
Methods of ordinary differential equations
58J05 - Elliptic equations on manifolds, general theory [See also 35-XX]
35A24 - Methods of ordinary differential equations
 

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