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arXiv · 1210.0782

A Reduction Method for Semilinear Elliptic Equations and Solutions Concentrating on Spheres

Filomena Pacella·P. N. Srikanth
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Abstract

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A in R^2m ;m >1, invariant by the action of a certain symmetry group can be reduced to a nonhomogenous similar problem in an annulus D in R^(m+1), invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A which concentrate on one or two (m-1) dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration tends to infinity.

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Filomena Pacella, P. N. Srikanth. 2012-10-02. A Reduction Method for Semilinear Elliptic Equations and Solutions Concentrating on Spheres. https://doi.org/10.1016/j.jfa.2014.03.004

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