Skip to main content
SpringerLink
Log in

Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We establish the solvability of certain linear nonhomogeneous equations and demonstrate that under reasonable technical conditions the convergence in \(L^{2}({\mathbb R}^{d})\) of their right sides implies the existence and the convergence in \(L^{2}({\mathbb R}^{d})\) of the solutions. In the first part of the work the equation involves the logarithmic Laplacian. In the second part we generalize the results derived by incorporating a shallow, short-range scalar potential into the problem. The argument relies on the methods of the spectral and scattering theory for the non-Fredholm Schrödinger type operators. As distinct from the preceding articles on the subject, for the operators involved in the equations the essential spectra fill the whole real line.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

All data generated or analyzed during this study are included in this published article. All of the material is owned by the authors and no permissions are required.

References

  1. Alfimov, G.L., Korobeinikov, A.S., Lustri, C.J., Pelinovsky, D.E.: Standing lattice solitons in the discrete NLS equation with saturation. Nonlinearity 32(9), 3445–3484 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amrouche, C., Girault, V., Giroire, J.: Dirichlet and neumann exterior problems for the \(n\)-dimensional Laplace operator: an approach in weighted Sobolev spaces. J. Math. Pures Appl. 76(1), 55–81 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amrouche, C., Bonzom, F.: Mixed exterior Laplace’s problem. J. Math. Anal. Appl. 338(1), 124–140 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bolley, P., Pham, T.L.: Propriétés d’indice en théorie höldérienne pour des opérateurs différentiels elliptiques dans \(R^n\). J. Math. Pures Appl. 72(1), 105–119 (1993)

    MathSciNet  MATH  Google Scholar 

  5. Bolley, P., Pham, T.L.: Propriété d’indice en théorie Höldérienne pour le problème extérieur de Dirichlet. Commun. Partial Differ. Equ. 26(1–2), 315–334 (2001)

    Article  MATH  Google Scholar 

  6. Benkirane, N.: Propriétés d’indice en théorie höldérienne pour des opérateurs elliptiques dans \(R^n\). C. R. Acad. Sci. Paris Ser. I Math. 307(11), 577–580 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Chen, H., Weth, T.: The Dirichlet problem for the logarithmic Laplacian. Commun. Partial Differ. Equ. 44(11), 1100–1139 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators with application to quantum mechanics and global geometry. Texts and monographs in physics, p. 319. Springer, Berlin (1987)

    MATH  Google Scholar 

  9. Ducrot, A., Marion, M., Volpert, V.: Systémes de réaction-diffusion sans propriété de Fredholm. C. R. Math. Acad. Sci. Paris 340(9), 659–664 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ducrot, A., Marion, M., Volpert, V.: Reaction-diffusion problems with non-Fredholm operators. Adv. Differ. Equ. 13(11–12), 1151–1192 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Efendiev, M. A.: Fredholm structures, topological invariants and applications. AIMS Series on Differential Equations & Dynamical Systems, 3. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2009), 205 pp

  12. Efendiev, M.A.: Finite and infinite dimensional attractors for evolution equations of mathematical physics: GAKUTO international series. Math. Sci. Appl. 33, 239 (2010)

    MATH  Google Scholar 

  13. Efendiev, M.A., Peletier, L.A.: On the large time behavior of solutions of fourth order parabolic equations and \(\varepsilon \)-entropy of their attractors. C. R. Math. Acad. Sci. Paris 344(2), 93–96 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Efendiev, M.A., Vougalter, V.: Solvability of some integro-differential equations with drift. Osaka J. Math. 57(2), 247–265 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Efendiev, M.A., Vougalter, V.: Solvability in the sense of sequences for some fourth order non-Fredholm operators. J. Differ. Equ. 271, 280–300 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Efendiev, M.A., Vougalter, V.: Existence of solutions for some non-Fredholm integro-differential equations with mixed diffusion. J. Differ. Equ. 284, 83–101 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Efendiev, M.A., Vougalter, V.: Linear and nonlinear non-Fredholm operators and their applications. Electron. Res. Arch. 30(2), 515–534 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  18. Efendiev, M.A., Vougalter, V.: Existence of solutions for some systems of integro-differential equations with transport and superdiffusion. Anal. Math. Phys. 12(5), 29 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  19. Efendiev, M.A., Zelik, S.V.: The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Commun. Pure Appl. Math. 54(6), 625–688 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gebran, H.G., Stuart, C.A.: Fredholm and properness properties of quasilinear elliptic systems of second order. Proc. Edinb. Math. Soc. 48(1), 91–124 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gebran, H.G., Stuart, C.A.: Exponential decay and Fredholm properties in second-order quasilinear elliptic systems. J. Differ. Equ. 249(1), 94–117 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Goldberg, M., Schlag, W.A.: A limiting absorption principle for the three-dimensional Schrödinger equation with \(L^{p}\) potentials. Int. Math. Res. Not. 75, 4049–4071 (2004)

    Article  MATH  Google Scholar 

  23. Jarohs, S., Saldana, A., Weth, T.: A new look at the fractional Poisson problem via the logarithmic Laplacian. J. Funct. Anal. 279(11), 108732 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jonsson, B.L.G., Merkli, M., Sigal, I.M., Ting, F.: Applied analysis. In preparation

  25. Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)

  26. Laptev, A., Weth, T.: Spectral properties of the logarithmic Laplacian. Anal. Math. Phys. 11(3), 24 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 278 pp, (1997)

  28. Rabier, P.J., Stuart, C.A.: Fredholm and properness properties of quasilinear elliptic operators on \({\mathbb{R} }^{N}\). Math. Nachr. 231, 129–168 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, III: Scattering Theory, p. 463. Academic Press, New York-London (1979)

    MATH  Google Scholar 

  30. Rodnianski, I., Schlag, W.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451–513 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Volpert, V.: Elliptic partial differential equations. Volume 1: Fredholm theory of elliptic problems in unbounded domains. Monographs in Mathematics, 101. Birkhäuser/Springer Basel AG, Basel, 639 pp, (2011)

  32. Volpert, V., Kazmierczak, B., Massot, M., Peradzynski, Z.: Solvability conditions for elliptic problems with non-Fredholm operators. Appl. Math. 29(2), 219–238 (2002)

    MathSciNet  MATH  Google Scholar 

  33. Volpert, V., Vougalter, V.: On the solvability conditions for a linearized Cahn-Hilliard equation. Rend. Istit. Mat. Univ. Trieste 43, 1–9 (2011)

    MathSciNet  MATH  Google Scholar 

  34. Volpert, V., Vougalter, V.: Solvability in the sense of sequences to some non-Fredholm operators. Electron. J. Differ. Equ. 160, 16 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Vougalter, V., Volpert, V.: On the solvability conditions for some non Fredholm operators. Int. J. Pure Appl. Math. 60(2), 169–191 (2010)

    MathSciNet  MATH  Google Scholar 

  36. Vougalter, V., Volpert, V.: Solvability conditions for some non-Fredholm operators. Proc. Edinb. Math. Soc. 54(1), 249–271 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Vougalter, V., Volpert, V.: On the solvability conditions for the diffusion equation with convection terms. Commun. Pure Appl. Anal. 11(1), 365–373 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Vougalter, V., Volpert, V.: Solvability conditions for a linearized Cahn-Hilliard equation of sixth order. Math. Model. Nat. Phenom. 7(2), 146–154 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Vougalter, V., Volpert, V.: Solvability conditions for some linear and nonlinear non-Fredholm elliptic problems. Anal. Math. Phys. 2(4), 473–496 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Vougalter, V., Volpert, V.: On the solvability in the sense of sequences for some non-Fredholm operators. Dyn. Partial Differ. Equ. 11(2), 109–124 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vougalter, V., Volpert, V.: Solvability in the sense of sequences for some non Fredholm operators related to the superdiffusion. J. Pseudo-Differ. Oper. Appl. 9(1), 25–46 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vougalter, V., Volpert, V.: On the solvability in the sense of sequences for some non-Fredholm operators related to the anomalous diffusion. Analysis of pseudo-differential operators, Trends Math., Birkhäuser/Springer, Cham, 229–257 (2019)

Download references

Funding

There was no funding received for this article.

Author information

Authors and Affiliations

  1. Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstrasse 1, 85764, Neuherberg, Germany

    Messoud Efendiev

  2. Department of Mathematics, Marmara University, Istanbul, Turkey

    Messoud Efendiev

  3. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Vitali Vougalter

Authors
  1. Messoud Efendiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vitali Vougalter
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The corresponding author has read the Springer journal policies on author responsibilities and submits this manuscript in accordance with those policies.

Corresponding author

Correspondence to Vitali Vougalter.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.

Ethical approval

The authors have read and understood the publishing policy, and submit this manuscript in accordance with this policy.

Aimal and human rights

The results of the work are applicable for both human and animal studies.

Consent for publication

The results in this manuscript have not been published elsewhere, nor are they under consideration by another publisher.

Additional information

Communicated by Adrian Constantin.

The article is dedicated to Professor Roald Sagdeev on the occasion of his 90th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Efendiev, M., Vougalter, V. Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian. Monatsh Math 202, 751–771 (2023). https://doi.org/10.1007/s00605-023-01834-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-023-01834-1

Keywords

Mathematics Subject Classification

Search

Navigation