Torus action on S^n and sign changing solutions for conformally invariant equations

Abstract

— We construct sequences of sign changing solutions for some conformally invariant semilinear elliptic equation which is defined in Sn, when n ≥ 4. The solutions we obtain have large energy and concentrate along some special submanifolds of Sn. For example, when n ≥ 4, we obtain sequences of solutions whose energy concentrates along one great circle or finitely many great circles which are linked (they correspond to Hopf links embedded in S3 × {0} ⊂ Sn). In dimension n ≥ 5, we obtain sequences of solutions whose energy concentrates along a two dimensional torus (which corresponds to a Clifford torus embedded in S3 × {0} ⊂ Sn).