Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schrödinger equation

Ao, Weiwei; Musso, Monica; Wei, Juncheng

doi:10.24033/bsmf.2774

Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schrödinger equation
[Non dégéneration de solutions non radiales qui changent de signe à l’équation non linéaire de Schrödinger]

Ao, Weiwei ¹ ; Musso, Monica ^2,³ ; Wei, Juncheng ⁴

¹ Department of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China
² Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
³ Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna 4860, Macul, Chile
⁴ Department of Mathematics, University of British Columbia, Vancouver, BC V6T1Z2, Canada

Bulletin de la Société Mathématique de France, Tome 147 (2019) no. 1, pp. 1-48

Abstract (VO)
Résumé

We prove that the non-radial sign-changing solutions to the nonlinear Schrödinger equation

Δ u - u + {| u |}^{p - 1} u = 0 in ℝ^{N}, u \in H^{1} (ℝ^{N})

constructed by Musso, Pacard, and Wei [16] are non-degenerate. This provides the first example of a non-degenerate sign-changing solution to the above nonlinear Schrödinger equation with finite energy.

Nous prouvons que les solutions non radiales qui changent de signe à l’équation non linéaire de Schrödinger

Δ u - u + {| u |}^{p - 1} u = 0 in ℝ^{N}, u \in H^{1} (ℝ^{N})

qui ont été construits par Musso, Pacard, Wei [16] sont non dégénérées. Ceci fournit le premier exemple de solutions qui changent de signe à l’équation non linéaire de Schrödinger avec énergie finie.

MR Zbl

DOI : 10.24033/bsmf.2774

Classification : 35B10, 92C40, 35B32
Keywords: Schrödinger equation, sign-changing solution, orthogonality condition
Mots-clés : Non-degeneracy, sign-changing solution, Schrodinger equation

Affiliations des auteurs :

Ao, Weiwei ¹ ; Musso, Monica ^{2, 3} ; Wei, Juncheng ⁴

¹ Department of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China
² Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
³ Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avda. Vicuña Mackenna 4860, Macul, Chile
⁴ Department of Mathematics, University of British Columbia, Vancouver, BC V6T1Z2, Canada

@article{BSMF_2019__147_1_1_0,
     author = {Ao, Weiwei and Musso, Monica and Wei, Juncheng},
     title = {Nondegeneracy of nonradial sign-changing solutions to the nonlinear {Schr\"odinger} equation},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {1--48},
     year = {2019},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {147},
     number = {1},
     doi = {10.24033/bsmf.2774},
     mrnumber = {3943737},
     zbl = {1437.35379},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2774/}
}

TY  - JOUR
AU  - Ao, Weiwei
AU  - Musso, Monica
AU  - Wei, Juncheng
TI  - Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schrödinger equation
JO  - Bulletin de la Société Mathématique de France
PY  - 2019
SP  - 1
EP  - 48
VL  - 147
IS  - 1
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/bsmf.2774/
DO  - 10.24033/bsmf.2774
LA  - en
ID  - BSMF_2019__147_1_1_0
ER  -

%0 Journal Article
%A Ao, Weiwei
%A Musso, Monica
%A Wei, Juncheng
%T Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schrödinger equation
%J Bulletin de la Société Mathématique de France
%D 2019
%P 1-48
%V 147
%N 1
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/bsmf.2774/
%R 10.24033/bsmf.2774
%G en
%F BSMF_2019__147_1_1_0

Ao, Weiwei; Musso, Monica; Wei, Juncheng. Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schrödinger equation. Bulletin de la Société Mathématique de France, Tome 147 (2019) no. 1, pp. 1-48. doi: 10.24033/bsmf.2774

Bibliographie
Cité par

Ao, W.W.; Musso, M.; Pacard, F.; Wei, J.C. Solutions without any symmetry for semilinear elliptic problems, J. Func. Anal., Volume 270 (2016), pp. 884-956 | MR | Zbl | DOI

Berestycki, H.; Lions, P.L. Nonlinear scalar field equations, II, Arch. Rat. Mech. Anal., Volume 82 (1981), pp. 347-375 | MR | Zbl | DOI

Berestycki, H.; Lions, P.L. Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., Volume 82 (1983), pp. 313-345 | MR | Zbl | DOI

Bartsch, T.; Willem, M. Infinitely many radial solutions of a semilinear elliptic problem on $R^{N}$ , Arch. Rational Mech. Anal., Volume 124 (1993), pp. 261-276 | MR | Zbl | DOI

Dancer, E.N. New solutions of equations on $R^{n}$ ., Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume 30 (2002), pp. 535-563 | MR | Zbl | Numdam

Duyckaerts, T.; Kenig, C.; Merle, F. Solutions of the focusing nonradial critical wave equation with the compactness property, Mathematics, Volume 15 (2014), pp. 731-808 | MR | Zbl

del Pino, M.; Kowalcyzk, M.; Pacard, F.; Wei, J. The Toda system and multiple-end solutions of autonomous planar elliptic problems, Advances in Mathematics, Volume 224 (2010), pp. 1462-1516 | MR | Zbl | DOI

del Pino, M.; Musso, M.; Pacard, F.; Pistoia, A. Large energy entire solutions for the Yamabe equation, J. Differential Equations, Volume 251 (2011), pp. 2568-2597 | MR | Zbl | DOI

del Pino, M.; Musso, M.; Pacard, F.; Pistoia, A. Torus action on $S^{n}$ and sign-changing solutions for conformally invariant equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 12 (2013), pp. 209-237 | MR | Zbl | Numdam

Gidas, B.; Ni, W.M.; Nirenberg, L. Symmetry and related properties via the maximun principle, Comm. Math. Phys., Volume 68 (1979), pp. 209-243 | MR | Zbl | DOI

Jleli, M.; Pacard, F. An end-to-end construction for compact constant mean curvature surfaces, Pacific Journal of Maths, Volume 221 (2005), pp. 81-108 | MR | Zbl | DOI

Kapouleas, N. Complete constant mean curvature surfaces in Euclidean three space, Ann. of Math., Volume 131 (1990), pp. 239-330 | MR | Zbl | DOI

Kapouleas, N. Compact constant mean curvature surfaces, J. Differential Geometry, Volume 33 (1991), pp. 683-715 | MR | Zbl | DOI

Kra, I.; Simanca, S. R. On Circulant Matrices, Notices. Amer. Math. Soc., Volume 59 (2012), pp. 368-377 | MR | Zbl | DOI

Kwong, M.K. Uniqueness of positive solutions of $Δ u - u + u^{p} = 0$ in $R^{n}$ , Arch. Rational Mech. Anal., Volume 105 (1989), pp. 243-266 | MR | Zbl | DOI

Kwong, M. K.; Zhang, L. Uniqueness of the positive solution of $Δ u + f (u) = 0$ in an annulus, Diff. Int. Eqns., Volume 4 (1991), pp. 583-599 | MR | Zbl

Lorca, S.; Ubilla, P. Symmetric and nonsymmetric solutions for an elliptic equation on $ℝ^{n}$ , Nonlinear Anal., Volume 58 (2004), pp. 961-968 | MR | Zbl | DOI

Malchiodi, A. Some new entire solutions of semilinear elliptic equations in $ℝ^{n}$ , Adv. Math., Volume 221 (2009), pp. 1843-1909 | MR | Zbl | DOI

Musso, M.; Pacard, F.; Wei, J.C. Finite energy sign changing solution with dihedral symmetry for the stationary non linear Schrödinger equation, Journal of European Mathematical Society, Volume 14 (2012), pp. 1923-1953 | MR | Zbl | DOI

Musso, M.; Wei, J.C. Nondegeneracy of nonradial nodal solutions to Yamabe problem, Comm. Math. Phys., Volume 340 (2015), pp. 1049-1107 | MR | Zbl | DOI

Ni, W. M.; Takagi, I. Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., Volume 70 (1993), pp. 247-281 | MR | Zbl

Struwe, M. Multiple solutions of differential equations without the Palais-Smale condition, Math. Ann., Volume 261 (1982), pp. 399-412 | MR | Zbl | DOI

Santra, S.; Wei, J. New entire positive solution for the nonlinear Schrödinger equation: coexistence of fronts and bumps, American J. Math., Volume 135 (2013), pp. 443-491 | MR | Zbl | DOI

Traizet, M. An embedded minimal surface with no symmetries, Journal of Diff. Geom., Volume 60 (2002), pp. 103-153 | MR | Zbl

Cité par Sources :