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

TL;DR

This paper introduces a reduction technique for semilinear elliptic equations with symmetry, enabling the construction of solutions concentrating on spheres and analyzing their Morse indices as parameters grow.

Contribution

It presents a novel reduction method transforming symmetric elliptic problems in annuli into simpler forms, facilitating the existence and index analysis of solutions.

Findings

Existence of solutions concentrating on spheres in symmetric annuli

Solutions' Morse indices tend to infinity with concentration parameter

Reduction method simplifies analysis of symmetric elliptic problems

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.