Skip to main content
SpringerLink
Log in

1D logistic reaction and p-Laplacian diffusion as p goes to one

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

Abstract

This work discusses the existence of the limit as p goes to 1 of the nontrivial solutions to the one-dimensional problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( |u_x|^{p-2} u_x\right) _x = \lambda |{u}|^{{p}-2}{u} -|{u}|^{{q}-2}{u}&{} \quad 0< x < 1\\ u(0)=u(1)=0, &{} \end{array}\right. } \end{aligned}$$

where \(\lambda \) is a positive parameter and the exponents p, q satisfy \(1< p < q\). We prove that solutions do converge to a limit function, which solves in a proper sense a Dirichlet problem involving the 1-Laplacian operator.

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

Fig. 1

Similar content being viewed by others

References

  1. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)

    MATH  Google Scholar 

  2. Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: Minimizing total variation flow. C. R. Acad. Sci. Paris Sér. I Math. 331(11), 867–872 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: The Dirichlet problem for the total variation flow. J. Funct. Anal. 180(2), 347–403 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. (4) 135(293–318), 1983 (1984)

    MathSciNet  Google Scholar 

  5. Cantrell, R.S., Cosner, C.: Spatial Ecology Via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology. Wiley, Chichester (2003)

    MATH  Google Scholar 

  6. Chang, K.C.: The spectrum of the 1-Laplace operator. Commun. Contemp. Math. 11(5), 865–894 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Crandall, M.G., Rabinowitz, P.H.: Nonlinear Sturm–Liouville eigenvalue problems and topological degree. J. Math. Mech. 19, 1083–1102 (1969/1970)

  8. de la Calle Ysern, B., Sabina de Lis, J.C., Segura de León, S.: The convective eigenvalues of the one-dimensional \(p\)-Laplacian as \(p\rightarrow 1\). J. Math. Anal. Appl. 484(1), 123738, 28 (2020)

    MATH  Google Scholar 

  9. del Pino, M., Elgueta, M., Manásevich, R.: A homotopic deformation along \(p\) of a Leray–Schauder degree result and existence for \((|u^{\prime }|^{p-2}u^{\prime })^{\prime }+f(t, u)=0,\;u(0)=u(T)=0,\;p>1\). J. Differ. Equ. 80(1), 1–13 (1989)

    MATH  Google Scholar 

  10. Demengel, F.: On some nonlinear partial differential equations involving the “1”-Laplacian and critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 4, 667–686 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Drábek, P.: Ranges of \(a\)-homogeneous operators and their perturbations. Časopis Pěst. Mat. 105(2), 167–183–208–209 (1980). With a loose Russian summary

    Article  MathSciNet  MATH  Google Scholar 

  12. Drábek, P.: Solvability and bifurcations of nonlinear equations, volume 264 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow; copublished in the United States with Wiley, New York (1992)

  13. Fife, P.C.: Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics, vol. 28. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  14. García-Melián, J.: Uniqueness for degenerate elliptic sublinear problems in the absence of dead cores. Electron. J. Differ. Equ. No. 110, 16 (2004)

  15. García-Melián, J., Rossi, J.D., Sabina de Lis, J.C.: Multiplicity of solutions to a nonlinear elliptic problem with nonlinear boundary conditions. NoDEA Nonlinear Differ. Equ. Appl. 21(3), 305–337 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. García-Melián, J., Sabina de Lis, J.: Stationary patterns to diffusion problems. Math. Methods Appl. Sci. 23(16), 1467–1489 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. García-Melián, J., Sabina de Lis, J.: Stationary profiles of degenerate problems when a parameter is large. Differ. Integral Equ. 13(10–12), 1201–1232 (2000)

    MathSciNet  MATH  Google Scholar 

  18. García Melián, J., Sabina de Lis, J.: Uniqueness to quasilinear problems for the \(p\)-Laplacian in radially symmetric domains. Nonlinear Anal. 43(7, Ser. A: Theory Methods), 803–835 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guedda, M., Véron, L.: Bifurcation phenomena associated to the \(p\)-Laplace operator. Trans. Am. Math. Soc. 310(1), 419–431 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Kamin, S., Véron, L.: Flat core properties associated to the \(p\)-Laplace operator. Proc. Am. Math. Soc. 118(4), 1079–1085 (1993)

    MathSciNet  MATH  Google Scholar 

  21. Latorre, M., Segura de León, S.: Existence and comparison results for an elliptic equation involving the 1-Laplacian and \(L^1\)-data. J. Evol. Equ. 18(1), 1–28 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Murray, J.D.: Mathematical Biology. I, Volume 17 of Interdisciplinary Applied Mathematics: An introduction, 3rd edn. Springer, New York (2002)

    Google Scholar 

  23. Ôtani, M.: A remark on certain nonlinear elliptic equations. Proc. Fac. Sci. Tokai Univ. 19, 23–28 (1984)

    MathSciNet  MATH  Google Scholar 

  24. Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  25. Reichel, W., Walter, W.: Radial solutions of equations and inequalities involving the \(p\)-Laplacian. J. Inequal. Appl. 1(1), 47–71 (1997)

    MathSciNet  MATH  Google Scholar 

  26. Sabina de Lis, J.C., Segura de León, S.: The limit as \(p\rightarrow 1\) of the higher eigenvalues of the \(p\)-Laplacian operator \(-\Delta _p\). Indiana Univ. Math. J. (2019) (to appear)

Download references

Acknowledgements

J. Sabina has been supported by DGI under Grant MTM2014-52822-P; S. Segura has been partially supported by MCIyU & FEDER, under project PGC2018-094775-B-I00.

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático and IUEA, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38203, La Laguna, Spain

    José Sabina Lis

  2. Departament d’Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100, Burjassot, València, Spain

    Sergio Segura de León

Authors
  1. José Sabina Lis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergio Segura de León
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José Sabina Lis.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sabina Lis, J., Segura de León, S. 1D logistic reaction and p-Laplacian diffusion as p goes to one. Ricerche mat 71, 529–547 (2022). https://doi.org/10.1007/s11587-020-00546-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-020-00546-0

Keywords

Mathematics Subject Classification

Search

Navigation