February  2021, 41(2): 701-746. doi: 10.3934/dcds.2020298

Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case

Univ. Sorbonne Paris Nord, Institut Galilée, LAGA, UMR 7539, 99 Avenue Jean Baptiste Clément, 93430 Villetaneuse, France

Received  March 2019 Revised  June 2020 Published  February 2021 Early access  August 2020

We consider the focusing $ L^2 $-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle $ \Theta \subset \mathbb{R}^3 $. We construct a solution behaving asymptotically as a solitary wave on $ \mathbb{R}^3, $ for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R. Killip, M. Visan and X. Zhang, which is the same as the one on the whole Euclidean space given by S. Roundenko and J. Holmer in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.

Citation: Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298
References:
[1]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.

[2]

M. D. BlairH. F. Smith and C. D. Sogge, Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., 354 (2012), 1397-1430.  doi: 10.1007/s00208-011-0772-y.

[3]

N. BurqP. Gérard and N. Tzvetkov, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal., 13 (2003), 1-19.  doi: 10.1007/s000390300000.

[4]

N. BurqP. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 295-318.  doi: 10.1016/j.anihpc.2003.03.002.

[5]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.

[6]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.

[7]

V. Combet, Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension, Discrete Contin. Dyn. Syst., 34 (2014), 1961-1993.  doi: 10.3934/dcds.2014.34.1961.

[8]

R. CôteY. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302.  doi: 10.4171/RMI/636.

[9]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.

[10]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840.  doi: 10.1007/s00039-009-0707-x.

[11]

T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3d cubic Schrödinger equation., Rev. Mat. Iberoam., 26 (2010), 1-56.  doi: 10.4171/RMI/592.

[12]

D. FangJ. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.  doi: 10.1007/s11425-011-4283-9.

[13]

N. Godet, Blow-up in several points for the nonlinear Schrödinger equation on a bounded domain, Differential Integral Equations, 24 (2011), 505-517. 

[14]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333.  doi: 10.1002/cpa.3160430302.

[15]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333.  doi: 10.1002/cpa.3160430302.

[16]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.

[17]

C. D. Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2014 (2014), 177-243. 

[18]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.

[19]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Analysis and PDE, 3 (2010), 261-293.  doi: 10.2140/apde.2010.3.261.

[20]

O. Ivanovici and G. Lebeau, Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples, C. R. Math. Acad. Sci. Paris, 355 (2017), 774-779.  doi: 10.1016/j.crma.2017.05.011.

[21]

O. Ivanovici and F. Planchon, On the energy critical Schrödinger equation in $3D$ non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1153-1177.  doi: 10.1016/j.anihpc.2010.04.001.

[22]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[23]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation., Acta Math., 201 (2008), 147-212.  doi: 10.1007/s11511-008-0031-6.

[24]

R. KillipM. Visan and X. Zhang, Riesz transforms outside a convex obstacle, International Mathematics Research Notices, 2016 (2016), 5875-5921.  doi: 10.1093/imrn/rnv338.

[25]

R. KillipM. Visan and X. Zhang, The focusing cubic NLS on exterior domains in three dimensions, Appl. Math. Res. Express. AMRX, 2016 (2016), 146-180.  doi: 10.1093/amrx/abv012.

[26]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[27]

D. Lafontaine, Strichartz estimates without loss outside two strictly convex obstacles, arXiv preprint, arXiv: 1709.03836, (2017).

[28]

D. Lafontaine, Strichartz estimates without loss outside many strictly convex obstacles, arXiv preprint, arXiv: 1811.12357, (2018).

[29]

D. LiH. Smith and X. Zhang, Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data, Math. Res. Lett., 19 (2012), 213-232.  doi: 10.4310/MRL.2012.v19.n1.a17.

[30]

Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864.  doi: 10.1016/j.anihpc.2006.01.001.

[31]

F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240.  doi: 10.1007/BF02096981.

[32]

F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261–290. doi: 10.24033/asens.2096.

[33]

F. Planchon and L. Vega, Scattering for solutions of NLS in the exterior of a 2D star-shaped obstacle, Math. Res. Lett., 19 (2012), 887-897.  doi: 10.4310/MRL.2012.v19.n4.a12.

[34]

W. Schlag, Spectral theory and nonlinear partial differential equations: A survey, Discrete Contin. Dyn. Syst., 15 (2006), 703-723.  doi: 10.3934/dcds.2006.15.703.

[35]

F. A. Shakra, On 2D nonlinear Schrödinger equation on non-trapping exterior domains, Rev. Mat. Iberoam., 31 (2015), 657-680.

[36]

T. Tao, Nonlinear Dispersive Equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. Local and global analysis. doi: 10.1090/cbms/106.

[37]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. 

[38]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

[39]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.

[40]

K. Yang, The focusing NLS on exterior domains in three dimensions, Commun. Pure Appl. Anal., 16 (2017), 2269-2297.  doi: 10.3934/cpaa.2017112.

show all references

References:
[1]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.

[2]

M. D. BlairH. F. Smith and C. D. Sogge, Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., 354 (2012), 1397-1430.  doi: 10.1007/s00208-011-0772-y.

[3]

N. BurqP. Gérard and N. Tzvetkov, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal., 13 (2003), 1-19.  doi: 10.1007/s000390300000.

[4]

N. BurqP. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 295-318.  doi: 10.1016/j.anihpc.2003.03.002.

[5]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.

[6]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.

[7]

V. Combet, Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension, Discrete Contin. Dyn. Syst., 34 (2014), 1961-1993.  doi: 10.3934/dcds.2014.34.1961.

[8]

R. CôteY. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302.  doi: 10.4171/RMI/636.

[9]

T. DuyckaertsJ. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250.  doi: 10.4310/MRL.2008.v15.n6.a13.

[10]

T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840.  doi: 10.1007/s00039-009-0707-x.

[11]

T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3d cubic Schrödinger equation., Rev. Mat. Iberoam., 26 (2010), 1-56.  doi: 10.4171/RMI/592.

[12]

D. FangJ. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062.  doi: 10.1007/s11425-011-4283-9.

[13]

N. Godet, Blow-up in several points for the nonlinear Schrödinger equation on a bounded domain, Differential Integral Equations, 24 (2011), 505-517. 

[14]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333.  doi: 10.1002/cpa.3160430302.

[15]

M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333.  doi: 10.1002/cpa.3160430302.

[16]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.

[17]

C. D. Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2014 (2014), 177-243. 

[18]

J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.

[19]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Analysis and PDE, 3 (2010), 261-293.  doi: 10.2140/apde.2010.3.261.

[20]

O. Ivanovici and G. Lebeau, Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples, C. R. Math. Acad. Sci. Paris, 355 (2017), 774-779.  doi: 10.1016/j.crma.2017.05.011.

[21]

O. Ivanovici and F. Planchon, On the energy critical Schrödinger equation in $3D$ non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1153-1177.  doi: 10.1016/j.anihpc.2010.04.001.

[22]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.

[23]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation., Acta Math., 201 (2008), 147-212.  doi: 10.1007/s11511-008-0031-6.

[24]

R. KillipM. Visan and X. Zhang, Riesz transforms outside a convex obstacle, International Mathematics Research Notices, 2016 (2016), 5875-5921.  doi: 10.1093/imrn/rnv338.

[25]

R. KillipM. Visan and X. Zhang, The focusing cubic NLS on exterior domains in three dimensions, Appl. Math. Res. Express. AMRX, 2016 (2016), 146-180.  doi: 10.1093/amrx/abv012.

[26]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[27]

D. Lafontaine, Strichartz estimates without loss outside two strictly convex obstacles, arXiv preprint, arXiv: 1709.03836, (2017).

[28]

D. Lafontaine, Strichartz estimates without loss outside many strictly convex obstacles, arXiv preprint, arXiv: 1811.12357, (2018).

[29]

D. LiH. Smith and X. Zhang, Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data, Math. Res. Lett., 19 (2012), 213-232.  doi: 10.4310/MRL.2012.v19.n1.a17.

[30]

Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864.  doi: 10.1016/j.anihpc.2006.01.001.

[31]

F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240.  doi: 10.1007/BF02096981.

[32]

F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261–290. doi: 10.24033/asens.2096.

[33]

F. Planchon and L. Vega, Scattering for solutions of NLS in the exterior of a 2D star-shaped obstacle, Math. Res. Lett., 19 (2012), 887-897.  doi: 10.4310/MRL.2012.v19.n4.a12.

[34]

W. Schlag, Spectral theory and nonlinear partial differential equations: A survey, Discrete Contin. Dyn. Syst., 15 (2006), 703-723.  doi: 10.3934/dcds.2006.15.703.

[35]

F. A. Shakra, On 2D nonlinear Schrödinger equation on non-trapping exterior domains, Rev. Mat. Iberoam., 31 (2015), 657-680.

[36]

T. Tao, Nonlinear Dispersive Equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. Local and global analysis. doi: 10.1090/cbms/106.

[37]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. 

[38]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.

[39]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.

[40]

K. Yang, The focusing NLS on exterior domains in three dimensions, Commun. Pure Appl. Anal., 16 (2017), 2269-2297.  doi: 10.3934/cpaa.2017112.

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