Singularly perturbed elliptic equations with solutions concentrating on a 1-dimensional orbit

Abstract

We consider a singularly perturbed elliptic equation with superlinear nonlinearity on an annulus in $R^4$, and look for solutions which are invariant under a fixed point free 1-parameter group action. We show that this problem can be reduced to a nonhomogeneous equation on a related annulus in dimension 3. The ground state solutions of this equation are single peak solutions which concentrate near the inner boundary. Transforming back, these solutions produce a family of solutions which concentrate along the orbit of the group action near the inner boundary of the domain.