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

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

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 $n \geq 4$ . The solutions we obtain have large energy and concentrate along some special submanifolds of $S^{n}$ . For example, for $n \geq 4$ 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 $S^{3} \times {0} \subset S^{n}$ ). In dimension $n \geq 5$ we obtain sequences of solutions whose energy concentrates along a two-dimensional torus (which corresponds to a Clifford torus embedded in $S^{3} \times {0} \subset S^{n}$ ).

Publié le : 2019-02-21

MR Zbl

Classification : 53C21, 35J65

@article{ASNSP_2013_5_12_1_209_0,
     author = {del Pino, Manuel and Musso, Monica and Pacard, Frank and Pistoia, Angela},
     title = {Torus action on $S^{n}$ and sign-changing solutions for conformally invariant equations},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {209--237},
     year = {2013},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {1},
     mrnumber = {3088442},
     zbl = {1267.53040},
     language = {en},
     url = {https://www.numdam.org/item/ASNSP_2013_5_12_1_209_0/}
}

TY  - JOUR
AU  - del Pino, Manuel
AU  - Musso, Monica
AU  - Pacard, Frank
AU  - Pistoia, Angela
TI  - Torus action on $S^{n}$ and sign-changing solutions for conformally invariant equations
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2013
SP  - 209
EP  - 237
VL  - 12
IS  - 1
PB  - Scuola Normale Superiore, Pisa
UR  - https://www.numdam.org/item/ASNSP_2013_5_12_1_209_0/
LA  - en
ID  - ASNSP_2013_5_12_1_209_0
ER  -

%0 Journal Article
%A del Pino, Manuel
%A Musso, Monica
%A Pacard, Frank
%A Pistoia, Angela
%T Torus action on $S^{n}$ and sign-changing solutions for conformally invariant equations
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2013
%P 209-237
%V 12
%N 1
%I Scuola Normale Superiore, Pisa
%U https://www.numdam.org/item/ASNSP_2013_5_12_1_209_0/
%G en
%F ASNSP_2013_5_12_1_209_0

del Pino, Manuel; Musso, Monica; Pacard, Frank; Pistoia, Angela. Torus action on $S^{n}$ and sign-changing solutions for conformally invariant equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 209-237. https://www.numdam.org/item/ASNSP_2013_5_12_1_209_0/

Bibliographie
Cité par

[1] A. Bahri and S. Chanillo, The difference of topology at infinity in changing- sign Yamabe problems on $S^{3}$ (the case of two masses). Comm. Pure Appl. Math. 54 (2001), 450–478. | MR | Zbl

[2] A. Bahri and Y. Xu, “Recent Progress in Conformal Geometry”, ICP Advanced Texts in Mathematics, 1, Imperial College Press, London, 2007. | MR | Zbl

[3] T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal. 117 (1993), 447–460. | MR | Zbl

[4] T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on $ℝ^{N}$ , Arch. Rational Mech. Anal. 124 (1993), 261–276. | MR | Zbl

[5] W. Ding, On a conformally invariant elliptic equation on $ℝ^{n}$ , Comm. Math. Phys. 107 (1986), 331–335. | MR | Zbl

[6] E. Hebey, “Introduction à l’analyse non linéaire sur les variétées”, Diderot éditeur, 1997. | Zbl

[7] M. Obata, Conformal changes of Riemannian metrics on a Euclidean sphere, In: “Differential Geometry” (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, 347–353. | MR | Zbl

[8] R. Schoen and S. T. Yau, “Lectures on Differential Geometry” Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. | MR | Zbl

[9] H.Y. Wang, The existence of nonminimal solutions to the Yang-Mills equation with group $S U (2)$ on $S^{2} \times S^{2}$ and $S^{1} \times S^{3}$ , J. Differential Geom. 34 (1991), 701–767. | MR | Zbl

[10] J. Wei and S. Yan, Infinitely many solutions for the prescribed scalar curvature problem on $S^{N}$ , J. Funct. Anal. 258 (2010), 3048–3081. | MR | Zbl