Existence of semi-nodal solutions for elliptic systems related to Gross-Pitaevskii equations
DOI:
https://doi.org/10.58997/ejde.2024.32Keywords:
Elliptic systems; variational methods; semi-nodal solutions; Gross-Pitaevskii equationAbstract
In this work we consider existence of semi-nodal solutions, i.e., solutions of the form \((u, v)\) with \(u>0\) and \(v^\pm:=\max\{0,\pm v\}\not\equiv0\) for a class of elliptic systems related to the Gross-Pitaevskii equation.
For more information see https://ejde.math.txstate.edu/Volumes/2024/32/abstr.html
References
A. Ambrosetti, E. Colorado; Standing waves of some coupled nonlinear Schrodinger equations. J. Lond. Math. Soc. (2) 75 (2007), no. 1, 67-82.
N. Akhmediev, A. Ankiewicz; Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.
T. Bartsch, N. Dancer, Z-Q, Wang; A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Differential Equations 37 (2010), no. 3-4, 345-361.
G. Cerami, S. Solimini, M. Struwe; Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. 69 (1986), no. 3, 289-306.
Z. Chen, C-S. Lin, W. Zou; Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations. J. Differential Equations 255 (2013), no. 11, 4289-4311.
Z. Chen, C.-S. Lin, W. Zou; Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrodinger system. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 859-897
Z. Chen, C.-S. Lin, W. Zou; Sign-changing solutions and phase separation for an elliptic system with critical exponent. Comm. Partial Differential Equations 39 (2014), no. 10, 1827- 1859.
Z. Chen, C.-S. Lin, W. Zou; Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrodinger system. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 859-897.
Z. Chen, C.-S. Lin, W. Zou; Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations. J. Differential Equations 255 (2013), no. 11, 4289-4311.
Z. Chen, W. Zou; An optimal constant for the existence of least energy solutions of a coupled Schrodinger system. Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 695-711.
M. Clapp, M. Soares; Energy estimates for seminodal solutions to an elliptic system with mixed couplings. NoDEA Nonlinear Differential Equations Appl. 30 (2023), no. 1, Paper No. 11, 33 pp.
I. Ekeland; On the variational principle, J. Anal. Appl. 17 (1974), 324-353.
B. Esry, C. Greene, J. Burke, J. Bohn; Hartree-Fock theory for double condesates, Phys. Rev. Lett., 78 (1997), 3594-3597.
D. J. Frantzeskakis; Dark solitons in atomic Bose-Einstein condesates: From theory to experiments. J. Phys. A (2010) 43:213001.
V. N. Ginzburg, A. A. Kochetkov, A. K. Potemkin, E.A. Khazanov; Suppression of smallscale self-focusing of high-power laser beams due to their self-filtration during propagation in free space Quantum Electron. 48 (2018) 325.
E. Khazanov, V. Ginzburg, A. Kochetkov; Self-Focusing Suppression in Ultrahigh-Intensity Lasers, 2018 Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR), Hong Kong, China, 2018, pp. 1-2.
Y. S. Kivshar, B. Luther-Davies; Dark optical solitons: physics and applications. Physics Reports (1998) 298:81-197.
T.-C. Lin, J. Wei; Ground state of N coupled nonlinear Schrodinger equations in Rn, n ¡Â 3. Comm. Math. Phys. 255 (2005), no. 3, 629-653.
Z. Liu, Z.-Q. Wang; Multiple bound states of nonlinear Schrodinger systems. Comm. Math. Phys. 282 (2008), no. 3, 721-731.
S. G. Lukishova, Y. V. Senatsky, N. E. Bykovsky, A. S. Scheulin; Beam Shaping and Suppression of Self-focusing in High-Peak-Power Nd:Glass Laser Systems Part of the book series: Topics in Applied Physics (TAP, volume 114, Chapter 8) DOI: 10.1007/978-0-387-34727-1 8
L. A. Maia, E. Montefusco, B. Pellacci; Positive solutions for a weakly coupled nonlinear Schrodinger system. J. Differential Equations 229 (2006), no. 2, 743-767.
C. Pethick, H. Smith; Bose-Einstein Condensation in Dilute Gases (2nd ed.). Cambridge: Cambridge University Press (2008).
B. Sirakov; Least energy solitary waves for a system of nonlinear Schrodinger equations in Rn. Comm. Math. Phys. 271 (2007), no. 1, 199-221.
G. Tarantello; On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincar¢¥e Anal. Non Lineaire 9 (1992), no. 3, 281-304.
J. Wei, T. Weth; Radial solutions and phase separation in a system of two coupled Schrodinger equations. Arch. Ration. Mech. Anal. 190 (2008), no. 1, 83-106.
Downloads
Published
License
Copyright (c) 2024 Joao Pablo Pinheiro da Silva, Edcarlos Domingos da Silva
This work is licensed under a Creative Commons Attribution 4.0 International License.
