Abstract
In this paper, we construct a family of new solutions for the following nonlinear Schrödinger system:
where P(y), Q(y) are positive radial potentials, μ > 0, v > 0 and \(\beta \in \mathbb{R}\). Motivated by the doubling construction of the entire finite energy sign-changing solution for the Yamabe equation in M. Medina and M. Musso (J. Math. Pures Appl. 2021), by using another type of building blocks, which are not equal to the ones adopted in S. Peng and Z.-Q. Wang (Arch. Ration. Mech. Anal. 2013), we successfully construct new segregated and synchronized vector solutions for the nonlinear Schrödinger system with more complex concentration structure. Our results extend the main results of S. Peng and Z.-Q. Wang (Arch. Ration. Mech. Anal. 2013), and in particular, for the segregated case, we well complement the previous works when the potentials P(y) and Q(y) decay in different rates.
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References
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris 342 (2006), 453–458.
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrodinger equations, J. Lond. Math. Soc. 75 (2007), 67–83.
W. Ao, L. Wang and W. Yao, Infinitely many solutions for nonlinear Schrodinger system with non-symmetric potentials, Commun. Pure Appl. Anal. 15 (2016), 965–989.
T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-proiri bounds and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations 37 (2010), 345–361.
T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl. 2 (2007), 353–367.
H. Berestycki, S. Terracini, K. Wang and J. Wei, On entire solutions of an elliptic system modeling phase seperations, Adv. Math. 243 (2013), 102–126.
Z. Chen, W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012), 515–551.
Z. Chen, W. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations 48 (2013), 695–711.
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differential Equations 52 (2015), 423–467.
M. del Pino, J Wei, W. Yao, Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials, Calc. Var. Partial Differential Equations 53 (2015), 473–523.
L. Duan and M. Musso, New type of solutions for the Nonlinear Schrödinger Equation in \({\mathbb{R}^N}\), J. Differential Equations 336 (2022), 479–504.
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397–108.
Y. Guo, B. Li and J. Wei, Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponent in \({\mathbb{R}^3}\), J. Differential Equations 256 (2014), 3463–3495.
Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear schrodinger systems, Comm. Math. Phys. 282 (2008), 721–731.
T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 403–439
T. Lin and J. Wei, Spikes in two-component system of nonlinear Schrodinger equations with trapping potential, J. Differential Equations 229 (2006), 538–569.
M. Kwong, Uniqueness of positive solutions of \(\Delta u - u + {u^p} = 0\,\,in\,{\mathbb{R}^3}\), Arch. Rational Mech. Anal. 105 (1989), 243–266.
F. Lin, W. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math. 60 (2007) 252–281.
W. Long, Z. Tang and S. Yang, Many synchronized vector solutions for a Bose-Einstein system, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 3293–3320.
M. Medina and M. Musso, Doubling nodal solution to the Yamabe equation in \({\mathbb{R}^n}\) with maximal rank, J. Math. Pures Appl. 152 (2021), 145–188.
S. Peng and Z.-Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrodinger systems, Arch. Rational. Mech. Anal. 208 (2013), 305–339.
S. Peng, Q. Wang and Z.-Q. Wang, On coupled nonlinear Schrodinger system with mixed couplings, Trans. Amer. Math. Soc. 371 (2019), 7559–7583.
B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}^n}\), Commum. Math. Phys. 271 (2007), 199–221.
J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrodinger equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 18 (2007), 279–293.
J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrodinger equations, Arch. Ration. Mech. Anal. 190 (2008), 83–106.
J. Wei and Y. Wu, Ground state of nonlinear Schrödinger system with mixed couplings,J. Math. Pures Appl. 141 (2020), 50–88.
J. Wei and S. Yan, Infinitely many positive solutions for the nonlinear Schrodinger equations in \({\mathbb{R}^3}\), Calc. Var. Partial Differential Equations 37 (2010), 423–439.
J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrodinger equations, Commun. Pure Appl. Anal. 11 (2012), 1003–1011.
Acknowledgements
Lipeng Duan was supported by NSFC (nos.11771167, 12201140), Technology Foundation of Guizhou Province ([2001]ZK008) and Guangdong Basic and Applied Basic Research Foundation (no.2022A1515111131). Xiao Luo was supported by the Anhui Provincial Natural Science Foundation (No.2308085MA05). Maoding Zhen was supported by the NSFC (no. 12201167) and the Fundamental Research Funds for the Central Universities (nos. JZ2023HGTB0218, JZ2022HGQA0155). We would like to thank Prof. L. Wang for his good suggestions on the proof of Theorem 1.2.
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Duan, L., Luo, X. & Zhen, M. New vector solutions for the cubic nonlinear schrödinger system. JAMA (2023). https://doi.org/10.1007/s11854-023-0315-y
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DOI: https://doi.org/10.1007/s11854-023-0315-y