Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-22T06:41:55.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiplicity and stability of normalized solutions to non-autonomous Schrödinger equation with mixed non-linearities

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  09 November 2023

Xinfu Li ,
Li Xu  and
Meiling Zhu
Xinfu Li
Affiliation:
School of Science, Tianjin University of Commerce, Tianjin, PR China (lxylxf@tjcu.edu.cn; beifang_xl@163.com)
Li Xu
Affiliation:
School of Science, Tianjin University of Commerce, Tianjin, PR China (lxylxf@tjcu.edu.cn; beifang_xl@163.com)
Meiling Zhu
Affiliation:
College of Computer Science and Engineering, Cangzhou Normal University, Cangzhou, Hebei, PR China (zhumeiling12365@163.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities

\begin{equation*}\begin{cases}-\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\\int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2,\end{cases}\end{equation*}
where $a, \epsilon, \eta \gt 0$, q is L2-subcritical, p is L2-supercritical, $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. The solutions obtained are local minimizers and probably not ground state solutions for the lack of symmetry of the potential h. Secondly, the stability of several different sets consisting of the local minimizers is analysed. Compared with the results of the corresponding autonomous equation, the appearance of the potential h increases the number of the local minimizers and the number of the stable sets. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$.

Keywords

normalized solutionsmultiplicitystabilitynon-autonomous Sobolev critical Schrödinger equationvariational methods

MSC classification

Primary: 35A15: Variational methods 35J10: Schrödinger operator 35B33: Critical exponents
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 1 , February 2024 , pp. 1 - 27
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alves, C. O., On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R}^N$, Z. Angew. Math. Phys. 73(3) (2022), .CrossRefGoogle Scholar
Alves, C. O., Ji, C. and Miyagaki, O. H., Multiplicity of normalized solutions for a Schrödinger equation with critical growth in $\mathbb{R}^N$. Preprint, arXiv:2103.07940, 2021.Google Scholar
Bartsch, T. and de Valeriola, S., Normalized solutions of nonlinear Schrödinger equations, Arch. Math. 100(1) (2013), 75–83.CrossRefGoogle Scholar
Bartsch, T. and Soave, N., A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal. 272(12) (2017), 4998–5037.CrossRefGoogle Scholar
Bartsch, T. and Soave, N., Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differential Equations 58(1) (2019), .CrossRefGoogle Scholar
Bellazzini, J., Boussaïd, N., Jeanjean, L. and Visciglia, N., Existence and stability of standing waves for supercritical NLS with a partial confinement, Commun. Math. Phys. 353(1) (2017), 229–251.CrossRefGoogle Scholar
Bellazzini, J., Jeanjean, L. and Luo, T., Existence and instability of standing waves with prescribed norm for a class of Schrödinger–Poisson equations, Proc. Lond. Math. Soc. 107(2) (2013), 303–339.CrossRefGoogle Scholar
Bieganowski, B. and Mederski, J., Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth, J. Funct. Anal. 280(11) (2011), .Google Scholar
Cao, D. M. and Noussair, E. S., Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R}^N$, Ann. Inst. Henri Poincaré 13(5) (1996), 567–588.CrossRefGoogle Scholar
Cazenave, T. and Lions, P. L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85(4) (1982), 549–561.CrossRefGoogle Scholar
Ding, Y. H. and Zhong, X. X., Normalized solution to the Schrödinger equation with potential and general nonlinear term: mass super-critical case, J. Differential Equations 334 (2022), 194–215.CrossRefGoogle Scholar
Gou, T. X. and Jeanjean, L., Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity 31(5) (2018), 2319–2345.CrossRefGoogle Scholar
Hajaiej, H. and Stuart, C. A., On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation, Adv. Nonlinear Stud. 4(4) (2004), 469–501.CrossRefGoogle Scholar
Hirata, J. and Tanaka, K., Nonlinear scalar field equations with constraint: mountain pass and symmetric mountain pass approaches, Adv. Nonlinear Stud. 19(2) (2019), 263–290.CrossRefGoogle Scholar
Ikoma, N. and Miyamoto, Y., Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities, Calc. Var. Partial Differential Equations 59(2) (2020), .CrossRefGoogle Scholar
Ikoma, N. and Tanaka, K., A note on deformation argument for normalized solutions of nonlinear Schrödinger equations and systems, Adv. Differential Equations 24(11-12) (2019), 609–646.CrossRefGoogle Scholar
Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28(10) (1997), 1633–1659.CrossRefGoogle Scholar
Jeanjean, L., Jendrej, J., Le, T. T. and Visciglia, N., Orbital stability of ground states for a Sobolev critical Schrödinger equation, J. Math. Pures Appl. 164 (2022), 158–179.CrossRefGoogle Scholar
Jeanjean, L. and Le, T. T., Multiple normalized solutions for a Sobolev critical Schrödinger equations, Math. Ann. 384(1-2) (2022), 101–134.CrossRefGoogle Scholar
Jeanjean, L. and Lu, S. S., Nonradial normalized solutions for nonlinear scalar field equations, Nonlinearity 32(12) (2019), 4942–4966.CrossRefGoogle Scholar
Jeanjean, L. and Lu, S. S., A mass supercritical problem revisited, Calc. Var. Partial Differential Equations 59(5) (2020), .CrossRefGoogle Scholar
Kang, J. C. and Tang, C. L., Normalized solutions for the nonlinear Schrödinger equation with potential and combined nonlinearities. Preprint, arXiv:2211.15354v2, 30 Nov, 2022.Google Scholar
Li, X., Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities, Calc. Var. Partial Differential Equations 60(5) (2021), .CrossRefGoogle Scholar
Li, X. and Zhao, J., Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential, Comput. Math. Appl. 79(2) (2020), 303–316.CrossRefGoogle Scholar
Luo, P., Peng, S. J. and Yan, S. S., Excited states on Bose-Einstein condensates with attractive interactions, Calc. Var. Partial Differential Equations 60(4) (2021), .CrossRefGoogle Scholar
Luo, X., Normalized standing waves for the Hartree equations, J. Differential Equations 267(7) (2019), 4493–4524.CrossRefGoogle Scholar
Luo, X., Wei, J., Yang, X. and Zhen, M., Normalized solutions for Schrödinger system with quadratic and cubic interactions, J. Differential Equations 314 (2022), 56–127.CrossRefGoogle Scholar
Luo, X., Yang, X. and Zou, W., Positive normalized solutions to nonlinear elliptic systems in $\mathbb{R}^4$ with critical Sobolev exponent. preprint, arXiv:2107.08708v1, 19 Jul, 2021.Google Scholar
Molle, R., Riey, G. and Verzini, G., Normalized solutions to mass supercritical Schrödinger equations with negative potential, J. Differential Equations 333 (2022), 302–331.CrossRefGoogle Scholar
Pellacci, B., Pistoia, A., Vaira, G. and Verzini, G., Normalized concentrating solutions to nonlinear elliptic problems, J. Differential Equations 275 (2021), 882–919.CrossRefGoogle Scholar
Peral Alonso, I., Multiplicity of Solutions for the p-laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations International Center for Theoretical Physics, Lecture Notes, Trieste (Italy) 1997.Google Scholar
Shibata, M., A new rearrangement inequality and its application for L 2-constraint minimizing problems, Math. Z. 287(1-2) (2017), 341–359.CrossRefGoogle Scholar
Soave, N., Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations 269(9) (2020), 6941–6987.CrossRefGoogle Scholar
Soave, N., Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal. 279(6) (2020), .CrossRefGoogle Scholar
Stefanov, A., On the normalized ground states of second order PDE’s with mixed power non-linearities, Commun. Math. Phys. 369(3) (2019), 929–971.CrossRefGoogle Scholar
Stuart, C. A., Bifurcation for Dirichlet problems without eigenvalues, Proc. Lond. Math. Soc. s3-45(1) (1982), 169–192.CrossRefGoogle Scholar
Tang, Z. W., Zhang, C. X., Zhang, L. Y. and Zhou, L. Y., Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well, Nonlinearity 35(8) (2022), 4624–4658.CrossRefGoogle Scholar
Tao, T., Visan, M. and Zhang, X., The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations 32(7-9) (2007), 1281–1343.CrossRefGoogle Scholar
Wei, J. C. and Wu, Y. Z., Normalized solutions of Schrödinger equations with critical Sobolev exponent and mixed nonlinearities, J. Funct. Anal. 283(6) (2022), .CrossRefGoogle Scholar
Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87(4) (1983), 567–576.CrossRefGoogle Scholar
Willem, M., Minimax Theorems (Birkhäuser, Boston, 1996).CrossRefGoogle Scholar
Yang, Z., Qi, S. J. and Zou, W. M., Normalized solutions of nonlinear Schrödinger equations with potentials and non-autonomous nonlinearities, J. Geom. Anal. 32(5) (2022), .CrossRefGoogle Scholar
Zhong, X. X. and Zou, W. M., A new deduction of the strict sub-additive inequality and its application: ground state normalized solution to Schrödinger equations with potential, Differential Integral Equations 36(1-2) (2023), 133–160.CrossRefGoogle Scholar