Nodal solutions for nonlinear Schrodinger systems
DOI:
https://doi.org/10.58997/ejde.2024.31Keywords:
Schrodinger system; sign-changing solutions; truncation method; method of invariant sets of descending flow.Abstract
In this article we consider the nonlinear Schrodinger system $$\displaylines{ - \Delta u_j + \lambda_j u_j = \sum_{i=1}^k \beta_{ij} u_i^2 u_j, \quad \hbox{in } \Omega, \cr u_j ( x ) = 0,\quad \hbox{on } \partial \Omega , \; j=1,l\dots,k , }$$ where \(\Omega\subset \mathbb{R}^N \) (\(N=2,3\)) is a bounded smooth domain, \(\lambda_j> 0\), \(j=1,\ldots,k\), \(\beta_{ij}\) are constants satisfying \(\beta_{jj}>0\), \(\beta_{ij}=\beta_{ji}\leq 0 \) for \(1\leq i< j\leq k\). The existence of sign-changing solutions is proved by the truncation method and the invariant sets of descending flow method.
For more information see https://ejde.math.txstate.edu/Volumes/2024/31/abstr.html
References
A. Ambrosetti, E. Colorado; Standing waves of some coupled nonlinear Schrodinger equations. J. Lond. Math. Soc., 75 (2007), 67-82.
T. Bartsch, Z. Liu, T. Weth; Sign-changing solutions of superlinear Schrodinger equations. Comm. Partial Differential Equations, 29 (2004), 25-42.
T. Bartsch, Z.-Q. Wang; Note on ground states of nonlinear Schrodinger systems, J. Patial Differential Equations, 19 (2006), 200-207.
S.-M. Chang, C.-S. Lin, T.-C. Lin, W.-W. Lin; Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Phys. D., 196 (2004), 341-361.
Z. Chen, C.-S. Lin, W. Zou; Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations. J. Differential Equations, 255 (2013), 4289-4311.
M. Contiand, L. Merizzi, S. Terracini; Remarks on variational methods and lower-upper solutions. Nonlinear Differential Equations Appl., 6 (1999) 371-393.
E. N. Dancer, J. Wei, T. Weth; A priori bounds versus multiple existence of positive solutions for a nonlinear Schrodinger system. Ann. Inst. H. poincarLe Anal. Non Lineaire, 27 (2010), 953-969.
B. D. Esry, C. H. Greene, J. P. Burke Jr, J. L. Bhon; Hartree-Fock theory for double condensates. Phys. Rev. Lett., 78 (1997), 3594-3597.
T.-C. Lin, J. Wei; Ground state of N coupled nonlinear Schrodinger equations in Rn, n . 3. Comm. Math. Phys., 255 (2005), 629-653.
J. Liu, X. Liu, Z.-Q. Wang; Multiple mixed states of nodal solutions for nonlinear Schrodinger systems. Calc. Var. Partial Differential Equations, 52 (2015), 565-586.
J. Liu, X. Liu, Z.-Q. Wang; Sign-changing solutions for coupled nonlinear Schrodinger equations with critical growth. J. Differential Equations, 261 (2016), 7194-7236.
X. Liu, J. Zhao; p-Laplacian equations in RN with finite potential via the truncation method. Adv. Nonlinear Stud., 17 (2017), 595-610.
Z. Liu, J. Sun; Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. J. Differential Equations, 172 (2001), 257-299.
Z. Liu, Z.-Q. Wang; Multiple bound states of nonlinear Schrodinger systems. Comm. Math. Phys., 282 (2008), 721-731.
Z. Liu, Z.-Q. Wang; Ground states and bound states of a nonlinear Schrodinger system. Adv. Nonlinear Stud., 10 (2010), 175-193.
E. Montefusco, B. Pellacci, M. Squassina; Semiclassical states for weakly coupled nonlinear Schrodinger systems. J. Eur. Math. Soc., 10 (2008), 47-71.
P. H. Rabinowitz; Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. Math. vol. 65. American Mathematical Soc., 1986.
E. Timmermans; Phase separation of Bose-Einstein condensates. Phys. Rev. Lett., 81 (1998), 5718-5721.
J.Wei, T.Weth; Radial solutions and phase separation in a system of two coupled Schrodinger equations. Arch. Ration Mech. Anal., 190 (2008), 83-106.
Downloads
Published
License
Copyright (c) 2024 Xue Zhou, Xiangqing Liu
This work is licensed under a Creative Commons Attribution 4.0 International License.