Chinese Annals of Mathematics, Series B

, Volume 38, Issue 1, pp 345–378 | Cite as

Flat solutions of some non-Lipschitz autonomous semilinear equations may be stable for N ≥ 3

  • Jesús Ildefonso Díaz
  • Jesús Hernández
  • Yavdat Il’yasov
Article
First Online:
Received:
Revised:

Abstract

The authors prove that flat ground state solutions (i.e. minimizing the energy and with gradient vanishing on the boundary of the domain) of the Dirichlet problem associated to some semilinear autonomous elliptic equations with a strong absorption term given by a non-Lipschitz function are unstable for dimensions N = 1,2 and they can be stable for N ≥ 3 for suitable values of the involved exponents.

Keywords

Semilinear elliptic and parabolic equation Strong absorption Spectral problem Nehari manifolds Pohozaev identity Flat solution Linearized stability Lyapunov function Global instability 

2000 MR Subject Classification

35J60 35J96 35R35 53C45 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Álvarez, L. and Díaz, J. I., On the retention of the interfaces in some elliptic and parabolic nonlinear problems, Discrete and Continuum Dynamical Systems, 25(1), 2009, 1–17.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Antontsev, S., Díaz, J. I. and Shmarev, S., Energy methods for free boundary problems, Applications to Nonlinear PDEs and Fluid Mechanics, Birkäuser, Boston, 2002.CrossRefMATHGoogle Scholar
  3. [3]
    Akagi, G. and Kajikiya, R., Stability of stationary solutions for semilinear heat equations with concave nonlinearity, Communications in Contemporary Mathematics, to appear.Google Scholar
  4. [4]
    Benilan, Ph., Brezis, H. and Crandall, M. G., A semilinear equation in L1(RN), Ann. Scuola Norm. Sup. Pisa, 4(2), 1975, 523–555.MathSciNetMATHGoogle Scholar
  5. [5]
    Bertsch, M. and Rostamian, R., The principle of linearized stability for a class of degenerate diffusion equations, J. Differ. Equat, 57, 1985, 373–405.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Bensoussan, A., Brezis, H. and Friedman, A., Estimates on the free boundary for quasi variational inequalities, Comm. PDEs, 2, 1977, 297–321.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Brezis, H., Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis, E. Zarantonello (ed.), Academic Press, New York, 1971, 101–156.Google Scholar
  8. [8]
    Brezis, H., Operateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espaces de Hilbert, North Holland, Amsterdam, 1973.MATHGoogle Scholar
  9. [9]
    Brezis, H., Solutions of variational inequalities with compact support, Uspekhi Mat. Nauk., 129, 1974, 103–108.MathSciNetMATHGoogle Scholar
  10. [10]
    Brezis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society, 88(3), 1983, 486–490.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Brezis, H. and Lieb, E., Minimum action solutions of some vector field equations, Comm. Math. Phys., 96, 1984, 97–113.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Brezis, H. and Nirenberg, L., Removable singularities for nonlinear elliptic equations, Topol. Methods Nonlinear Anal., 9, 1997, 201–219.MathSciNetMATHGoogle Scholar
  13. [13]
    Brezis, H. and Friedman, A., Estimates on the support of solutions of parabolic variational inequalities, Illinois J. Math., 20, 1976, 82–97.MathSciNetMATHGoogle Scholar
  14. [14]
    Cazenave, T., Dickstein, T. and Escobedo, M., A semilinear heat equation with concave-convex nonlinearity, Rendiconti di Matematica, Serie VII, 19, 1999, 211–242.MathSciNetMATHGoogle Scholar
  15. [15]
    Cazenave, T. and Haraux, A., An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, New York, 1998.MATHGoogle Scholar
  16. [16]
    Cortázar, C., Elgueta, M. and Felmer, P., Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation, Comm. PDEs, 21, 1996, 507–520.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Cortázar, C., Elgueta, M. and Felmer, P., On a semi-linear elliptic problem in RNwith a non-Lipschitzian non-linearity, Advances in Diff. Eqs., 1, 1996, 199–218.MATHGoogle Scholar
  18. [18]
    Daners, D. and Koch Medina, P., Abstract evolution equations, periodic problems and applications, Pitman Research Notes in Mathematics Series, Vol. 279, Longman, Harlow, Essex, 1992.MATHGoogle Scholar
  19. [19]
    Dao, A. N., Díaz, J. I. and Sauvy, P., Quenching phenomenon of singular parabolic problems with L1 initial data, Electronic J. Diff. Eqs., 2016(136), 2016, 1–16.MATHGoogle Scholar
  20. [20]
    Dávila, J. and Montenegro, M., Existence and asymptotic behavior for a singular parabolic equation, Transactions of the AMS, 357, 2005, 1801–1828MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Díaz, J. I., Nonlinear Partial Differential Equations and Free Boundaries, Pitman Research Notes in Mathematics Series, Vol. 106, Pitman, London, 1985.MATHGoogle Scholar
  22. [22]
    Díaz, J. I., On the Haïm Brezis pioneering contributions on the location of free boundaries, Proceedings of the Fifth European Conference on Elliptic and Parabolic Problems; A special tribute to the work of Haïm Brezis, M. Chipot et al. (eds.), Birkhauser Verlag, Bassel, 2005, 217–234.Google Scholar
  23. [23]
    Díaz, J. I., On the ambiguous treatment of the Schrödinger equation for the infinite potential well and an alternative via flat solutions: The one-dimensional case, Interfaces and Free Boundaries, 17, 2015, 333–351.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    Díaz, J. I., On the ambiguous treatment of the Schrödinger equation for infinite potential well and an alternative via flat solutions: The multi-dimensional case, to appear.Google Scholar
  25. [25]
    Díaz, J. I. and Hernández, J., Global bifurcation and continua of non-negative solutions for a quasilinear elliptic problem, C. R. Acad. Sci. Paris, 329, 1999, 587–592.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Díaz, J. I. and Hernández, J., Positive and nodal solutions bifurcating from the infinity for a semilinear equation: Solutions with compact support, Portugaliae Math., 72(2), 2015, 145–160.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    Díaz, J. I., Hernández, J. and Ilyasov, Y., On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption, Nonlinear Analysis Series A: Theory, Mehods and Applications, 119, 2015, 484–500.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    Díaz, J. I., Hernández, J. and Maagli, H., in preparation.Google Scholar
  29. [29]
    Díaz, J. I., Hernández, J. and Mancebo, F. J., Branches of positive and free boundary solutions for some singular quasilinear elliptic problems, J. Math. Anal. Appl., 352, 2009, 449–474.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    Díaz, J. I., Mingazzini, T. and Ramos, A. M., On the optimal control for a semilinear equation with cost depending on the free boundary, Networks and Heterogeneous Media, 7, 2012, 605–615.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    Díaz, J. I. and Tello, L., On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, L, 1999, 19–51.Google Scholar
  32. [32]
    Díaz, J. I. and Vrabie, I. I., Existence for reaction-diffusion systems, A compactness method approach, J. Math. Anal. Appl., 188, 1994, 521–540.Google Scholar
  33. [33]
    Giacomoni, J., Sauvy, P. and Shmarev, S., Complete quenching for a quasilinear parabolic equation, J. Math. Anal. Appl., 410, 2014, 607–624.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    Lions, P. L., The concentration-compactness principle in the calculus of variations, The locally compact case, Part 2, Annales de l’Institut Henri Poincaré: Analyse Non Linèaire, 1(4), 1984, 223–283.Google Scholar
  35. [35]
    Dickstein, F., On semilinear parabolic problems with non-Lipschitz nonlinearity, to appear.Google Scholar
  36. [36]
    Fujita, H. and Watanabe, S., On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations, Comm. Pure Appl. Math., 21, 1968, 631–652.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Oder, 2nd ed., Springer-Verlag, Berlin, 1983.CrossRefMATHGoogle Scholar
  38. [38]
    Hernández, J., Mancebo, F. J. and Vega, J. M., On the linearization ofsome singular nonlinear elliptic problems and applications, Annales de l’Institut Henri Poincaré: Analyse Non Linèaire, 19, 2002, 777–813.MATHGoogle Scholar
  39. [39]
    Il’yasov, Y. S., Nonlocal investigations of bifurcations of solutions of nonlinear elliptic equations, Izv. Math., 66(6), 2002, 1103–1130.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    Il’yasov, Y. S., On calculation of the bifurcations by the fibering approach, Harmonic, Wavelet and P-adic Analysis, N. M. Chuong, et al. (eds.), World Scientific Publishing, Singapore, 2007, 141–155.Google Scholar
  41. [41]
    Il’yasov, Y. S., On critical exponent for an elliptic equation with non-Lipschitz nonlinearity, Dynamical Systems, Supplement, 2011, 698–706.MATHGoogle Scholar
  42. [42]
    Il’yasov, Y. S. and Egorov, Y., Höpf maximum principle violation for elliptic equations with non-Lipschitz nonlinearity, Nonlin. Anal., 72, 2010, 3346–3355.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    Il’yasov, Y. S. and Takac, P., Optimal-regularity, Pohozhaev’s identity, and nonexistence of weak solutions to some quasilinear elliptic equations, Journal of Differential Equations, 252(3), 2012, 2792–2822.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    Kaper, H. and Kwong, M., Free boundary problems for Emden-Fowler equation, Differential and Integral Equations, 3, 1990, 353–362.MathSciNetMATHGoogle Scholar
  45. [45]
    Kaper, H., Kwong, M. and Li, Y., Symmetry results for reaction-diffusion equations, Differential and Integral Equations, 6, 1993, 1045–1056.MathSciNetMATHGoogle Scholar
  46. [46]
    Ozolins, V., Lai, R., Caflisch, R. and Osher, S., Compressed modes for variational problems in mathematics and physics, Proc. Natl. Acad. Sci. USA, 110(46), 2013, 18368–18373.MathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    Ozolins, V., Lai, R., Caflisch, R. and Osher, S., Compressed plane waves yield a compactly supported multiresolution basis for the Laplace operator, Proc. Natl. Acad. Sci. USA, 111(5), 2014, 1691–1696.MathSciNetCrossRefGoogle Scholar
  48. [48]
    Payne, L. E. and Sattinger, D. H., Saddle points and instability of nonlinear hyperbolic equations, Israel Journal of Mathematics, 22(3–4), 1975, 273–303.MathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    Pohozaev, S. I., Eigenfunctions of the equation u+f(u) = 0, Sov. Math. Doklady, 5, 1965, 1408–1411.MathSciNetMATHGoogle Scholar
  50. [50]
    Pohozaev, S. I., On the method of fibering a solution in nonlinear boundary value problems, Proc. Stekl. Ins. Math., 192, 1990, 157–173.MathSciNetGoogle Scholar
  51. [51]
    Serrin, J. and Zhou, H., Symmetry of ground states of quasilinear elliptic equations, Archive for Rational Mechanics and Analysis, 148(4), 1999, 265–290.MathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    Szulkin, A. and Weth, T., The method of Nehari manifold, Handbook of nonconvex analysis and applications, D. Y. Gao et al. (ed.), International Press, Somerville, MA,2010, 597–632.Google Scholar
  53. [53]
    Struwe, M., Variational Methods, Application to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1996.MATHGoogle Scholar
  54. [54]
    Vrabie, I. I., Compactness Methods for Nonlinear Evolutions, Pitman Longman, London, 1987.MATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Jesús Ildefonso Díaz
    • 1
  • Jesús Hernández
    • 2
  • Yavdat Il’yasov
    • 3
  1. 1.Instituto de Matemática InterdisciplinarUniversidad Complutense de MadridMadridSpain
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridCantoblanco, MadridSpain
  3. 3.Institute of MathematicsUfa Science Center of RAS 112UfaRussia

Personalised recommendations