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Asymptotic behavior of boundary blow-up solutions to elliptic equations

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Abstract

This paper is concerned with the asymptotic behavior on \({\partial\Omega}\) of boundary blow-up solutions to semilinear elliptic equations

$$\left\{\begin{array}{ll} \Delta u=b(x)f(u),~~ &x\in \Omega, \\ u(x)=\infty, ~~ &x\in\partial\Omega,\end{array} \right.$$

where b(x) is a nonnegative function on \({\Omega}\) and may vanish on \({\partial\Omega}\) at a very degenerate rate; f is nonnegative function on [0,∞) and normalized regularly varying or rapidly varying at infinity. The main feature of this paper is to establish a unified and explicit asymptotic formula when the function f is normalized regularly varying or grows faster than any power function at infinity. The effect of the mean curvature of the nearest point on the boundary in the second-order approximation of the boundary blow-up solution is also discussed. Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.

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References

  1. Alarcón S., Díaz G., Rey J.M.: The influence of sources terms on the boundary behavior of the large solutions of quasilinear elliptic equations: the power like case. Z. Angew. Math. Phys. 64, 659–677 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alarcón S., Díaz G., Letelier R., Rey J.M.: Expanding the asymptotic explosive boundary behavior of large solutions to a semilinear elliptic equation. Nonlinear Anal. 72, 2426–2443 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anedda C., Porru G.: Estimates for boundary blow-up solutions of semilinear elliptic equations. Int. J. Math. Anal. 2, 213–234 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Anedda C., Porru G.: Boundary behaviour for solutions of boundary blow-up problems in a borderline case. J. Math. Anal. Appl. 352, 35–47 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bandle C., Marcus M.: On second order effects in the boundary behaviour of large solutions of semilinear elliptic problems. Differ. Integral Equ. 11 23–34 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Bandle C.: Asymptotic behaviour of large solutions of quasilinear elliptic problems. Z. Angew. Math. Phys. 54, 731–738 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bandle C., Marcus M.: Dependence of blowup rate of large solutions of semilinear elliptic equations, on the curvature of the boundary. Complex Var. Theory Appl. 49, 555–570 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bingham N.H., Goldie C.M., Teugels J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)

    Book  MATH  Google Scholar 

  9. Cano-Casanova S., López-Gómez J.: Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line. J. Differ. Equ. 244, 3180–3203 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cano-Casanova S., López-Gómez J.: Blow-up rates of radially symmetric large solutions. J. Math. Anal. Appl. 352, 166–174 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cîrstea F.: Elliptic equations with competing rapidly varying nonlinearities and boundary blow-up. Adv. Differ. Equ. 12, 995–1030 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Cîrstea F., Rădulescu V.: Existence and uniqueness of blow-up solutions for a class of logistic equations. Commun. Contemp. Math. 4, 559–586 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cîrstea F., Rădulescu V.: Asymptotics for the blow-up boundary solution of the logistic equation with absorption. C. R. Acad. Sci. Paris Ser. I. 336, 231–236 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cîrstea F., Rădulescu V.: Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach. Asymptot. Anal. 46, 275–298 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Cîrstea F., Rădulescu V.: Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type. Trans. Am. Math. Soc. 359, 3275–3286 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Costin O., Dupaigne L.: Boundary blow-up solutions in the unit ball: asymptotics, uniqueness and symmetry. J. Differ. Equ. 249, 931–964 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Díaz J.I., Lazzo M., Schmidt P.G.: Large solutions for a system of elliptic equations arising from fluid dynamics. SIAM J. Math. Anal. 37, 490–513 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Du Y., Huang Q.: Blow-up solutions for a class of semilinear elliptic and parabolic equations. SIAM J. Math. Anal. 31, 1–18 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Du Y., Yamada Y.: On the long-time limit of positive solutions to the degenerate logistic equation. Discrete Contin. Dyn. Syst. A. 25, 123–132 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dumont S., Dupaigne L., Goubet O., Rădulescu V.: Back to the Keller–Osserman condition for boundary blow-up solutions. Adv. Nonlinear Stud. 7, 271–298 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. García-Melián J.: Boundary behavior for large solutions of elliptic equations with singular weights. Nonlinear Anal. 67, 818–826 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. García-Melián J.: Uniqueness of positive solutions for a boundary value problem. J. Math. Anal. Appl. 360, 530–536 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. García-Melián J., Albornoz R.L., Sabinade Lis J.: Uniqueness and asymptotic behavior for solutions of semilinear problems with boundary blow-up. Proc. Am. Math. Soc. 129, 3593–3602 (2001)

    Article  MATH  Google Scholar 

  24. Geluk J.L., de Haan L.: Regular Variation, Extensions and Tauberian Theorems. CWI Tract/Centrum Wisk, Inform., Amsterdam (1987)

    MATH  Google Scholar 

  25. Gómez-Meñasco R., López-Gómez J.: On the existence and numerical computation of classical and nonclassical solutions for a family of elliptic boundary value problems. Nonlinear Anal. 48, 567–605 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Huang S., Tian Q.: Second order estimates for large solutions of elliptic equations. Nonlinear Anal. 74, 2031–2044 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huang S., Tian Q., Zhang S., Xi J., Fan Z.: The exact blow-up rates of large solutions for semilinear elliptic equations. Nonlinear Anal. 73, 3489–3501 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Huang S., Tian Q., Zhang S., Xi J.: A second-order estimate for blow-up solutions of elliptic equations. Nonlinear Anal. 74, 2342–2350 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Huang S., Li W-T., Wang M.: A unified asymptotic behavior of boundary blow-up solutions to elliptic equations. Differ. Integral Equ. 26, 675–692 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Lazer A.C., McKenna P.J.: Asymptotic behaviour of solutions of boundary blow up problems. Differ. Integral Equ. 7, 1001–1019 (1994)

    MathSciNet  MATH  Google Scholar 

  31. Li H., Pang P.Y.H., Wang M.: Boundary blow-up solutions for logistic-type porous media equations with nonregular source. J. Lond. Math. Soc. 80, 273–294 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. López-Gómez J.: Dynamics of parabolic equations: from classical solutions to metasolutions. Differ. Integral Equ. 16, 813–828 (2003)

    MathSciNet  MATH  Google Scholar 

  33. López-Gómez J.: The boundary blow-up rate of large solutions. J. Differ. Equ. 195, 25–45 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. López-Gómez, J.: Uniqueness of large solutions for a class of radially symmetric elliptic equations. In: Cano-Casanova, S., López-Gómez, J., Mora-Corral, C. (eds.) Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, World Sci. Publ., Hackensack, NJ, Singapore, pp. 75–110 (2005)

  35. López-Gómez J.: Optimal uniqueness theorems and exact blow-up rates of large solutions. J. Differ. Equ. 224, 385–439 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. López-Gómez, J.: Uniqueness of radially symmetric large solutions. Discrete Contin. Dyn. Syst. In: Proceedings of the 6th AIMS Conference, Poitiers (France), pp. 677–686 (2007)

  37. Mi L., Liu B.: Second order expansion for blowup solutions of semilinear elliptic problems. Nonlinear Anal. 75, 2591–2613 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. del Pino M., Letelier R.: The influence of domain geometry in boundary blow-up elliptic problems. Nonlinear Anal. 48, 897–904 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rădulescu, V.: Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities. In: Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. IV, pp. 485–593, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2007)

  40. Repovš D.: Asymptotics for singular solutions of quasilinear elliptic equations with an absorption term. J. Math. Anal. Appl. 395, 78–85 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Resnick S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  42. Seneta E.: Regularly Varying Functions. In: Lecture Notes in Math. Vol. 508, Springer, Berlin (1976)

  43. Xie Z.: Uniqueness and blow-up rate of large solutions for elliptic equation \({-\Delta u=\lambda u-b(x)f(u)}\). J. Differ. Equ. 247, 344–363 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang Z.: The second expansion of the solution for a singular elliptic boundary value problem. J. Math. Anal. Appl. 381, 922–934 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang Z.: The second expansion of large solutions for semilinear elliptic equations. Nonlinear Anal. 74, 3445–3457 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang Z.: The existence and boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms. Adv. Nonlinear Anal. 3, 165–185 (2014)

    MathSciNet  MATH  Google Scholar 

  47. Zhang Z.: Boundary behavior of large solutions to the Monge-Ampère equations with weights. J. Differ. Equ. 259, 2080–2100 (2015)

    Article  MATH  Google Scholar 

  48. Zhang Z., Ma Y., Mi L., Li X.: Blow-up rates of large solutions for elliptic equations. J. Differ. Equ. 249, 180–199 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Mathematics and Computer, Northwest University for Nationalities, Lanzhou, 730000, Gansu, People’s Republic of China

    Shuibo Huang

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Huang, S. Asymptotic behavior of boundary blow-up solutions to elliptic equations. Z. Angew. Math. Phys. 67, 3 (2016). https://doi.org/10.1007/s00033-015-0606-y

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  • DOI: https://doi.org/10.1007/s00033-015-0606-y

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