Torus action on S n and sign-changing solutions for conformally invariant equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 1, p. 209-237

We construct sequences of sign-changing solutions for some conformally invariant semilinear elliptic equation which is defined S n , when n4. The solutions we obtain have large energy and concentrate along some special submanifolds of S n . For example, for n4 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 ×{0}S n ). In dimension n5 we obtain sequences of solutions whose energy concentrates along a two-dimensional torus (which corresponds to a Clifford torus embedded in S 3 ×{0}S n ).

Published online : 2019-02-21
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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {1},
     year = {2013},
     pages = {209-237},
     zbl = {1267.53040},
     mrnumber = {3088442},
     language = {en},
     url = {http://www.numdam.org/item/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, Serie 5, Volume 12 (2013) no. 1, pp. 209-237. http://www.numdam.org/item/ASNSP_2013_5_12_1_209_0/

[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 1808650 | Zbl 1035.53050

[2] A. Bahri and Y. Xu, “Recent Progress in Conformal Geometry”, ICP Advanced Texts in Mathematics, 1, Imperial College Press, London, 2007. | MR 2323753 | Zbl 1128.53002

[3] T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal. 117 (1993), 447–460. | MR 1244943 | Zbl 0790.35021

[4] T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on N , Arch. Rational Mech. Anal. 124 (1993), 261–276. | MR 1237913 | Zbl 0790.35020

[5] W. Ding, On a conformally invariant elliptic equation on n , Comm. Math. Phys. 107 (1986), 331–335. | MR 863646 | Zbl 0608.35017

[6] E. Hebey, “Introduction à l’analyse non linéaire sur les variétées”, Diderot éditeur, 1997. | Zbl 0918.58001

[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 324585 | Zbl 0243.53038

[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 1333601 | Zbl 0830.53001

[9] H.Y. Wang, The existence of nonminimal solutions to the Yang-Mills equation with group SU(2) on S 2 ×S 2 and S 1 ×S 3 , J. Differential Geom. 34 (1991), 701–767. | MR 1139645 | Zbl 0744.53011

[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 2595734 | Zbl 1209.53028