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

del Pino, Manuel ; Musso, Monica  ; Pacard, Frank ; Pistoia, Angela 

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 209-237

Résumé

We construct sequences of sign-changing solutions for some conformally invariant semilinear elliptic equation which is defined $S^n$, when $n\ge 4$. The solutions we obtain have large energy and concentrate along some special submanifolds of $S^n$. For example, for $n\ge 4$ we obtain sequences of solutions whose energy concentrates along one great circle or finitely many great circles which are linked to each other (and they correspond to Hopf links embedded in $S^3\times \left\{0\right\}\subset S^n$). In dimension $n\ge 5$ we obtain sequences of solutions whose energy concentrates along a two-dimensional torus (which corresponds to a Clifford torus embedded in $S^3\times \left\{0\right\}\subset S^n$).

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Publié le : 2019-02-21

Classification : 53C21, 35J65