Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T15:59:22.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic equation with van der Waals type potential

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  18 October 2022

Yu Su  and
Senli Liu
Yu Su
Affiliation:
School of Mathematics and Big data, Anhui University of Science and Technology, Huainan, Anhui 232001, China (yusumath@aust.edu.cn)
Senli Liu
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China (mathliusl@csu.edu.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the Lieb's translation lemma in Coulomb–Sobolev space and then apply it to investigate the existence of Pohožaev type ground state solution for elliptic equation with van der Waals type potential.

Keywords

Coulomb–Sobolev spaceelliptic equationground stateLieb's translation lemmaPohožaev manifold

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 4 , November 2022 , pp. 1048 - 1062
Check for updates
Copyright
© The Author(s), 2022. 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

Bellazzini, J., Frank, R. and Visciglia, N., Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems, Math. Ann. 360 (2014), 653–673.CrossRefGoogle Scholar
Bellazzini, J., Ghimenti, M. and Ozawa, T., Sharp lower bounds for Coulomb energy, Math. Res. Lett. 23(3) (2016), 621–632.CrossRefGoogle Scholar
Bellazzini, J., Ghimenti, M., Mercuri, C., Moroz, V. and Van Schaftingen, J., Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces, Trans. Amer. Math. Soc. 370(11) (2018), 8285–8310.CrossRefGoogle Scholar
Brézis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 8 (1983), 486–490.CrossRefGoogle Scholar
Cao, D., Jia, H. and Luo, X., Standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials, J. Differ. Equ. 276(9) (2021), 228–263.CrossRefGoogle Scholar
Dzyaloshinskii, I., Lifshitz, E. and Pitaevskii, L., The general theory of van der Waals’ forces, Adv. Phys. 10 (1961), 165–209.CrossRefGoogle Scholar
Fröhlich, J., Lieb, E. and Loss, M., Stability of Coulomb systems with magnetic fields. I. The one-electron atom, Comm. Math. Phys. 104 (1986), 251–270.CrossRefGoogle Scholar
Ianni, I. and Ruiz, D., Ground and bound states for a static Schrödinger-Poisson-Slater problem, Commun. Contemp. Math. 14 (2012), 1250003.CrossRefGoogle Scholar
Lieb, E., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74 (1983), 441–448.CrossRefGoogle Scholar
Lieb, E. and Loss, M., Analysis, Graduate Studies in Mathematics, Volume 14 (American Mathematical Society, Providence, 2001).CrossRefGoogle Scholar
Mercuri, C., Moroz, V. and Van Schaftingen, J., Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency, Calc. Var. Partial Differ. Equ. 55 (2016), 146.CrossRefGoogle Scholar
Ruiz, D., On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198(1) (2010), 349–368.CrossRefGoogle Scholar
Willem, M., Minimax theorems (Birkhäuser, Boston, 1996).CrossRefGoogle Scholar
Yang, D., Li, P. and Tang, K., The ground state van der Waals potentials of the calcium dimer and calcium rare-gas complexes, J. Chem. Phys. 131 (2009), 154301.CrossRefGoogle Scholar
Yang, J. and Yu, W., Schrödinger equations with van der Waals type potentials, J. Math. Anal. Appl. 471 (2019), 267–298.CrossRefGoogle Scholar
Zheng, Y. and Narayanaswamy, A., Lifshitz theory of van der Waals pressure in dissipative media, Phys. Rev. A 83 (2011), 042504.CrossRefGoogle Scholar