Existence, Multiplicity and Stability of Normalized Solutions to Non-Autonomous Schrödinger Equation with Mixed Nonlinearities

Abstract

This paper studies the existence, multiplicity and stability of normalized solutions to the following non-autonomous Schrödinger equation with mixed nonlinearities

$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+W(\epsilon x)u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u,\quad x\in \mathbb {R}^N, \\ \int _{\mathbb {R}^N}|u|^2dx=a^2, \end{array}\right. } \end{aligned}$

where $N\ge 1$, $a, \epsilon , \mu >0$, $\textit{q}$ is $L^2$-subcritical, $\textit{p}$ is $L^2$-supercritical, $\lambda \in \mathbb {R}$ is an unknown parameter that appears as a Lagrange multiplier, $\textit{W}$ is a bounded and continuous function. The existence and multiplicity of normalized solutions to the above equation are studied by using different methods. It is proved that the numbers of normalized solutions are at least the numbers of global minimum points of $\textit{W}$ when $\epsilon $ is small enough. In particular, our results cover the Sobolev critical case $p=\frac{2N}{N-2} (N\ge 3)$.

Keywords

Normalized solutions / Multiplicity / Sobolev critical exponent / Non-autonomous Schrödinger equation / Stability / Mixed nonlinearities / 35A15 / 35J10 / 35B33

References

[1] AlvesCO. On existence of multiple normalized solutions to a class of elliptic problems in whole $\mathbb{R} ^N$. Z. Angew. Math. Phys., 2022, 7397

[2] AlvesCO, JiC. Normalized solutions for the Schrödinger equations with $L^2$-subcritical growth and different types of potentials. J. Geom. Anal., 2022, 32165

[3] Alves, C.O., Ji, C., Miyagaki, O.H.: Multiplicity of normalized solutions for a Schrödinger equation with critical growth in $\mathbb{R}^N$, arXiv:2103.07940, 2021

[4] BartschT, de ValeriolaS. Normalized solutions of nonlinear Schrödinger equations. Arch. Math., 2013, 100: 75-83

[5] BartschT, SoaveN. A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. J. Funct. Anal., 2017, 272: 4998-5037

[6] BellazziniJ, BoussaïdN, JeanjeanL, ViscigliaN. Existence and stability of standing waves for supercritical NLS with a partial confinement. Commun. Math. Phys., 2017, 353: 229-251

[7] BellazziniJ, JeanjeanL, LuoT. Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations. Proc. Lond. Math. Soc., 2013, 107(2): 303-339

[8] BieganowskiB, MederskiJ. Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth. J. Funct. Anal., 2011, 28011108989

[9] CaoD, NoussairES. Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R} ^N$. Ann. Inst. Henri Poincaré, 1996, 13(5): 567-588

[10] Cazenave, T.: Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol.10, New York Uni-versity, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, Providence, RI, ISBN0-8218-3399-5, xiv+323 (2003)

[11] CazenaveT, LionsPL. Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys., 1982, 85(4): 549-561

[12] DingY, ZhongX. Normalized solution to the Schrödinger equation with potential and general nonlinear term: mass super-critical case. J. Differ. Eq., 2022, 334: 194-215

[13] HajaiejH, StuartCA. On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation. Adv. Nonlinear Stud., 2004, 4(4): 469-501

[14] HirataJ, TanakaK. Nonlinear scalar field equations with constraint: Mountain pass and symmetric mountain pass approaches. Adv. Nonlinear Stud., 2019, 19(2): 263-290

[15] IkomaN, MiyamotoY. Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities. Calc. Var. Partial Diff. Equ., 2020, 59248

[16] IkomaN, TanakaK. A note on deformation argument for normalized solutions of nonlinear Schrödinger equations and systems. Adv. Diff. Equa., 2019, 24(11–12): 609-646

[17] JeanjeanL. Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal., 1997, 28: 1633-1659

[18] JeanjeanL, JendrejJ, LeTT, ViscigliaN. Orbital stability of ground states for a Sobolev critical Schrödinger equation. J. de Mathématiques Pures et Appliquées, 2022, 164: 158-179

[19] Jeanjean, L., Le, T.T.: Multiple normalized solutions for a Sobolev critical Schrödinger equations. Math. Ann. 384, 101–134 (2022) https://doi.org/10.1007/s00208-021-02228-0

[20] JeanjeanL, LuS. Nonradial normalized solutions for nonlinear scalar field equations. Nonlinearity, 2019, 32(12): 4942-4966

[21] JeanjeanL, LuS. A mass supercritical problem revisited. Calc. Var. Part. Differ. Equ., 2020, 59174

[22] Kang, J., Tang, C.: Normalized solutions for the nonlinear Schrödinger equation with potential and combined nonlinearities, Nonlinear Analysis 246, 113581 (2024)

[23] LiX. Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities. Calc. Var. Part. Differ. Equ., 2021, 60169

[24] Li, X., Xu, L., Zhu, M.: Multiplicity and orbital stability of normalized solutions to non-autonomous Schrodinger equation with mixed nonlinearities, Proceedings of the Edinburgh Mathematical Society 67(1), 1–27 (2024)

[25] LiX, ZhaoJ. Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential. Comput. Math. Appl., 2020, 79(2): 303-316

[26] LuoP, PengS, YanS. Excited states on Bose-Einstein condensates with attractive interactions. Calc. Var. Part. Differ. Equ., 2021, 60155

[27] MolleR, RieyG, VerziniG. Normalized solutions to mass supercritical Schrödinger equations with negative potential. J. Differ. Eq., 2022, 333: 302-331

[28] PellacciB, PistoiaA, VairaG, VerziniG. Normalized concentrating solutions to nonlinear elliptic problems. J. Differ. Eq., 2021, 275: 882-919

[29] Peral Alonso, I.: Multiplicity of solutions for the p-laplacian, Second School of Nonlinear Functional Analysis and Applications to Differential Equations, Trieste, (1997)

[30] ShibataM. A new rearrangement inequality and its application for $L^2$-constraint minimizing problems. Math. Z., 2017, 287: 341-359

[31] SoaveN. Normalized ground states for the NLS equation with combined nonlinearities. J. Differ. Eq., 2020, 269(9): 6941-6987

[32] SoaveN. Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case. J. Funct. Anal., 2020, 2796108610

[33] StefanovA. On the normalized ground states of second order PDE’s with mixed power non-linearities. Commun. Math. Phys., 2019, 369(3): 929-971

[34] StuartCA. Bifurcation for Dirichlet problems without eigenvalues. Proc. Lond. Math. Soc., 1982, 45: 169-192

[35] TangZ, ZhangC, ZhangL, ZhouL. Normalized multibump solutions to nonlinear Schrödinger equations with steep potential well. Nonlinearity, 2022, 35: 4624-4658

[36] WeiJ, WuY. Normalized solutions of Schrödinger equations with critical Sobolev exponent and mixed nonlinearities. J. Funct. Anal., 2022, 2836109574

[37] WeinsteinMI. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 1983, 87: 567-576

[38] WillemMMinimax theorems, 1996, Boston. Birkhäuser.

[39] YangZ, QiS, ZouW. Normalized solutions of nonlinear Schrödinger equations with potentials and non-autonomous nonlinearities. J. Geom. Anal., 2022, 32159

[40] ZhongX, ZouW. A new deduction of the strict sub-additive inequality and its application: Ground state normalized solution to Schrödinger equations with potential. Differ. Integral Eq., 2023, 36: 133-160

Funding

National Natural Science Foundation of China(12001403)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.