Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential

Fangyuan Dong; Fangyuan Dong

doi:10.3934/cam.2024023

Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 487-508. doi: 10.3934/cam.2024023

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Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential

Fangyuan Dong ^,

School of Mathematics and Physics, University of Science and Technology Beijing, Xueyuan Road 30, Haidian, Beijing 100083, China

Received: 29 November 2023 Revised: 19 January 2024 Accepted: 06 March 2024 Published: 05 July 2024
35Q55, 35A15, 35J60, 35B09

In this article, we mainly study the global existence of multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb type potential
$ \begin{equation*} -\Delta u+V(\epsilon x) u = \lambda (I_\alpha * |u|^p)|u|^{p-1}+u \log u^2 \text { in } \mathbb{R}^3, \end{equation*} $
where $ u \in H^1(\mathbb{R}^3) $, $ \epsilon > 0 $, $ V $ is a continuous function with a global minimum, and Coulomb type energies with $ 0 < \alpha < 3 $ and $ p \geq 1 $. We explore the existence of local positive solutions without the functional having to be a combination of a $ C^1 $ functional and a convex semicontinuous functional, as is required in the global case.
- variational method,
- logarithmic Schrödinger equation,
- multiple solutions,
- Coulomb potential
Citation: Fangyuan Dong. Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential[J]. Communications in Analysis and Mechanics, 2024, 16(3): 487-508. doi: 10.3934/cam.2024023

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Abstract

In this article, we mainly study the global existence of multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb type potential

$ \begin{equation*} -\Delta u+V(\epsilon x) u = \lambda (I_\alpha * |u|^p)|u|^{p-1}+u \log u^2 \text { in } \mathbb{R}^3, \end{equation*} $

where $ u \in H^1(\mathbb{R}^3) $, $ \epsilon > 0 $, $ V $ is a continuous function with a global minimum, and Coulomb type energies with $ 0 < \alpha < 3 $ and $ p \geq 1 $. We explore the existence of local positive solutions without the functional having to be a combination of a $ C^1 $ functional and a convex semicontinuous functional, as is required in the global case.

References

[1]	T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127–1140. https://doi.org/10.1016/0362-546X(83)90022-6 doi: 10.1016/0362-546X(83)90022-6
[2]	T. Cazenave, An introduction to nonlinear Schrödinger equations, Universidade federal do Rio de Janeiro, Centro de ciências matemáticas e da natureza, Instituto de matemática, 1989.
[3]	T. Cazenave, A. Haraux, Équations d'évolution avec non linéarité logarithmique, Annales de la Faculté des sciences de Toulouse: Mathématiques, 2 (1980), 21–51. https://doi.org/10.5802/afst.543 doi: 10.5802/afst.543
[4]	I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics, Ann Phys-New York, 100 (1976), 62–93. https://doi.org/10.1016/0003-4916(76)90057-9
[5]	I. Bialynicki-Birula, J. Mycielski, Gaussons: solitons of the logarithmic Schrödinger equation, Phys Scripta, 20 (1979), 539. https://doi.org/10.1088/0031-8949/20/3-4/033 doi: 10.1088/0031-8949/20/3-4/033
[6]	I. Bialynicki-Birula, J. Mycielski, Wave equations with logarithmic nonlinearities, Bull. Acad. Pol. Sci. Cl., 3 (1975), 461–466.
[7]	W. Lian, M. S. Ahmed, R. Xu, Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239–257. https://doi.org/10.1016/j.na.2019.02.015 doi: 10.1016/j.na.2019.02.015
[8]	X. Wang, Y. Chen, Y. Yang, J. Li, R. Xu, Kirchhoff-type system with linear weak damping and logarithmic nonlinearities, Nonlinear Anal., 188 (2019), 475–499. https://doi.org/10.1016/j.na.2019.06.019 doi: 10.1016/j.na.2019.06.019
[9]	Y. Chen, R. Xu, Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity, Nonlinear Anal., 192 (2020), 111664. https://doi.org/10.1016/j.na.2019.111664 doi: 10.1016/j.na.2019.111664
[10]	R. Carles, C. Su, Numerical study of the logarithmic Schrödinger equation with repulsive harmonic potential, Discrete Contin. Dyn. Syst. Ser. B, 28 (2023), 3136–3159. https://doi.org/10.3934/dcdsb.2022206 doi: 10.3934/dcdsb.2022206
[11]	E. H. Lieb, M. Loss, Analysis, volume 14. American Mathematical Soc., 2001. https://doi.org/10.1090/gsm/014
[12]	C. O. Alves, D. C. de Morais Filho, Existence and concentration of positive solutions for a Schrödinger logarithmic equation, Z. Angew. Math. Phys., 69 (2018), 1–22. https://doi.org/10.1007/s00033-018-1038-2 doi: 10.1007/s00033-018-1038-2
[13]	C. O. Alves, C. Ji, Multiple positive solutions for a Schrödinger logarithmic equation, Discrete Cont Dyn-A, preprint, arXiv: 1901.10329.
[14]	L. Guo, Y. Sun, G. Shi, Ground states for fractional nonlocal equations with logarithmic nonlinearity, Opusc. Math., 42 (2022), 157–178. https://doi.org/10.7494/OpMath.2022.42.2.157 doi: 10.7494/OpMath.2022.42.2.157
[15]	I. Bachar, H. Mli, H. Eltayeb, Nonnegative solutions for a class of semipositone nonlinear elliptic equations in bounded domains of $\mathbb R^n$, Opusc. Math., 42, 2022. https://doi.org/10.7494/OpMath.2022.42.6.793 doi: 10.7494/OpMath.2022.42.6.793
[16]	A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann I H Poincare-An, 3 (1986), 77–109. https://doi.org/10.1016/S0294-1449(16)30389-4 doi: 10.1016/S0294-1449(16)30389-4
[17]	B. Ilan, G. Fibich, G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM. J. Appl. Math., 62 (2002), 1437–1462. https://doi.org/10.1137/S0036139901387241 doi: 10.1137/S0036139901387241
[18]	B. Pausader, S. Shao, The mass-critical fourth-order Schrödinger equation in high dimensions, J Hyperbolic Differ Eq, 7 (2010), 651–705. https://doi.org/10.1142/S0219891610002256 doi: 10.1142/S0219891610002256
[19]	H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, P. Am. Math. Soc., 88 (1983), 486–490. https://doi.org/10.1090/S0002-9939-1983-0699419-3 doi: 10.1090/S0002-9939-1983-0699419-3
[20]	C. Ji, A. Szulkin, A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241–254. https://doi.org/10.1016/j.jmaa.2015.11.071 doi: 10.1016/j.jmaa.2015.11.071
[21]	M. Squassina, A. Szulkin, Multiple solutions to logarithmic Schrödinger equations with periodic potential, Calc. Var. Partial Dif., 54 (2015), 585–597. https://doi.org/10.1007/s00526-014-0796-8 doi: 10.1007/s00526-014-0796-8
[22]	M. Willem, Minimax theorems, volume 24, Springer Science & Business Media, 1997. https://doi.org/10.1007/978-1-4612-4146-1
[23]	D. Cao, E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $r^n$, Ann I H Poincare-An, 13 (1996), 567–588. https://doi.org/10.1016/S0294-1449(16)30115-9 doi: 10.1016/S0294-1449(16)30115-9
[24]	C. Mercuri, V. Moroz, J Van Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger–poisson–slater equations at the critical frequency, Calc. Var. Partial Dif., 55 (2016), 1–58. https://doi.org/10.1007/s00526-016-1079-3 doi: 10.1007/s00526-016-1079-3

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