Existence of normalized solutions for the Schrödinger equation

Shengbing Deng; Qiaoran Wu; Shengbing Deng; Qiaoran Wu

doi:10.3934/cam.2023028

Communications in Analysis and Mechanics

2023, Volume 15, Issue 3: 575-585. doi: 10.3934/cam.2023028

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Existence of normalized solutions for the Schrödinger equation

Shengbing Deng ^,,
Qiaoran Wu

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, P.R. China

Received: 20 July 2023 Revised: 26 August 2023 Accepted: 29 August 2023 Published: 18 September 2023
35J15, 35J20, 35J91

In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities.

$ \begin{align*} &\left\{\begin{array}{ll} -\Delta u = \lambda u+\mu\lvert u \rvert^{q-2}u+\lvert u \rvert^{p-2}u&{\mbox{in}}\ \mathbb{R}^N,\\ \int_{\mathbb{R}^N}\lvert u\rvert^2dx = a^2, \end{array}\right. \end{align*} $

where $ N\geqslant 3 $, $ 2 < q < 2+\frac{4}{N} $, $ p = 2^* = \frac{2N}{N-2} $, $ a, \mu > 0 $ and $ \lambda\in\mathbb{R} $ is a Lagrange multiplier. Since the existence result for $ 2+\frac{4}{N} < p < 2^* $ has been proved, using an approximation method, that is let $ p\rightarrow 2^* $, we obtain that there exists a mountain-pass type solution for $ p = 2^* $.
- normalized solutions,
- Schrödinger equation,
- Sobolev critical nonlinearities,
- approximation method,
- mountain-pass type solution
Citation: Shengbing Deng, Qiaoran Wu. Existence of normalized solutions for the Schrödinger equation[J]. Communications in Analysis and Mechanics, 2023, 15(3): 575-585. doi: 10.3934/cam.2023028

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Abstract

In this paper, we devote to studying the existence of normalized solutions for the following Schrödinger equation with Sobolev critical nonlinearities.

$ \begin{align*} &\left\{\begin{array}{ll} -\Delta u = \lambda u+\mu\lvert u \rvert^{q-2}u+\lvert u \rvert^{p-2}u&{\mbox{in}}\ \mathbb{R}^N,\\ \int_{\mathbb{R}^N}\lvert u\rvert^2dx = a^2, \end{array}\right. \end{align*} $

where $ N\geqslant 3 $, $ 2 < q < 2+\frac{4}{N} $, $ p = 2^* = \frac{2N}{N-2} $, $ a, \mu > 0 $ and $ \lambda\in\mathbb{R} $ is a Lagrange multiplier. Since the existence result for $ 2+\frac{4}{N} < p < 2^* $ has been proved, using an approximation method, that is let $ p\rightarrow 2^* $, we obtain that there exists a mountain-pass type solution for $ p = 2^* $.

References

[1]	M. B. Benboubker, H. Benkhalou, H. Hjiaj, I. Nyanquini, Entropy solutions for elliptic Schrödinger type equations under Fourier boundary conditions, Rend. Circ. Mat. Palermo (2), 72 (2023), 2831–2855. https://doi.org/10.1007/s12215-022-00822-y doi: 10.1007/s12215-022-00822-y
[2]	T. Cazenave, F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s(\mathbb{R}^N)$, Nonlinear Anal., 14 (1990), 807–836. https://doi.org/10.1016/0362-546X(90)90023-A doi: 10.1016/0362-546X(90)90023-A
[3]	M. Khiddi, L. Essafi, Infinitely many solutions for quasilinear Schrödinger equations with sign-changing nonlinearity without the aid of 4-superlinear at infinity, Demonstr. Math., 55 (2022), 831–842. https://doi.org/10.1515/dema-2022-0169 doi: 10.1515/dema-2022-0169
[4]	T. Tao, M. Visan, X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281–1343. https://doi.org/10.1080/03605300701588805 doi: 10.1080/03605300701588805
[5]	L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633–1659. https://doi.org/10.1016/S0362-546X(96)00021-1 doi: 10.1016/S0362-546X(96)00021-1
[6]	L. Jeanjean, T. T. Le, Multiple normalized solutions for a Sobolev critical Schrödinger equation, Math. Ann., 384 (2022), 101–134. https://doi.org/10.1007/s00208-021-02228-0 doi: 10.1007/s00208-021-02228-0
[7]	X. Li, Existence of normalized ground states for the Sobolev critical Schrödinger equation with combined nonlinearities, Calc. Var. Partial Differential Equations, 60 (2021). https://doi.org/10.1007/s00526-021-02020-7 doi: 10.1007/s00526-021-02020-7
[8]	P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223–283. http://www.numdam.org/item?id=AIHPC_1984__1_4_223_0
[9]	N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations, 269 (2020), 6941–6987. https://doi.org/10.1016/j.jde.2020.05.016 doi: 10.1016/j.jde.2020.05.016
[10]	N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 269 (2020), 6941–6987. https://doi.org/10.1016/j.jfa.2020.108610 doi: 10.1016/j.jfa.2020.108610
[11]	J. Wei, Y. Wu, Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities, J. Funct. Anal., 283 (2022). https://doi.org/10.1016/j.jfa.2022.109574 doi: 10.1016/j.jfa.2022.109574
[12]	H. Luo, Z. Zhang, Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differential Equations, 59 (2020). https://doi.org/10.1007/s00526-020-01814-5 doi: 10.1007/s00526-020-01814-5
[13]	M. Zhen, B. Zhang, V. D. Radulescu, Normalized solutions for nonlinear coupled fractional systems: low and high perturbations in the attractive case, Discrete Contin. Dyn. Syst., 41 (2021), 2653–2676. https://doi.org/10.3934/dcds.2020379 doi: 10.3934/dcds.2020379
[14]	J. Zuo, C. Liu, C. Vetro, Normalized solutions to the fractional Schrödinger equation with potential, Mediterr. J. Math., 20 (2023). https://doi.org/10.1007/s00009-023-02422-1 doi: 10.1007/s00009-023-02422-1
[15]	T. Bartsch, N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998–5037. https://doi.org/10.1016/j.jfa.2017.01.025 doi: 10.1016/j.jfa.2017.01.025
[16]	M. Li, J. He, H. Xu, Yang, M. Yang, Normalized solutions for a coupled fractional Schrödinger system in low dimensions, Bound. Value Probl., (2020), 1687–2762. https://doi.org/10.1186/s13661-020-01463-9 doi: 10.1186/s13661-020-01463-9
[17]	M. Liu, W. Zou, Normalized solutions for a system of fractional Schrödinger equations with linear coupling, Minimax Theory Appl., 7 (2022), 303–320.
[18]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87 (1982), 567–576. http://projecteuclid.org/euclid.cmp/1103922134
[19]	C. A. Stuart, Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc., 45 (1982), 169–192. https://doi.org/10.1112/plms/s3-45.1.169 doi: 10.1112/plms/s3-45.1.169
[20]	H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437–477. https://doi.org/10.1002/cpa.3160360405 doi: 10.1002/cpa.3160360405
[21]	G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4), 110 (1976), 353–372. https://doi.org/10.1007/BF02418013 doi: 10.1007/BF02418013
[22]	H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486–490. https://doi.org/10.2307/2044999 doi: 10.2307/2044999

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