Hopf reduction and orbit concentrating solutions for a class of superlinear elliptic equations

We consider singularly perturbed equations of the form{ε2Δu−u+up=0 in A⊂RN,u>0 in A,u=0 on ∂A, where A is a annulus and p>1. It has been conjectured for a long time that such problems possess solutions having m-dimensional concentration sets for every 0≤m≤N−1. For N=3 solutions with 2-dimensional and 0-dimensional concentration sets are known, while no result was available for 1-dimensional concentration sets. We answer positively this conjecture, proving the existence of solutions which concentrate on circles S1. The proof relies on work by Santra-Wei who proved the existence of solutions concentrating on a Clifford torus S1×S1 for an annulus in R4. We extend this result to equations with weights. Then we use the Hopf fibration to show that these solutions give rise to the S1-concentrating solutions in R3.