Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

BOUNDARY BEHAVIOR OF LARGE SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS

  • Sun, Juan (School of Mathematical Sciences, Nanjing Normal University) ;
  • Yang, Zuodong (Institute of Mathematics, School of Mathematics Science, Nanjing Normal University)
  • Received : 2009.12.10
  • Accepted : 2010.10.23
  • Published : 2011.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, our main purpose is to consider the quasilinear elliptic equation $$div(|{\nabla}u|^{p-2}{\nabla}u)=(p-1)f(u)$$ on a bounded smooth domain ${\Omega}\;{\subset}\;R^N$, where p > 1, N > 1 and f is a smooth, increasing function in [0, ${\infty}$). We get some estimates of a solution u satisfying $u(x){\rightarrow}{\infty}$ as $d(x,\;{\partial}{\Omega}){\rightarrow}0$ under different conditions on f.

Keywords

References

  1. Ahmed Mohammed, Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations, J.Math.Anal.Appl.298(2004), 621-637. https://doi.org/10.1016/j.jmaa.2004.05.030
  2. J.B.Keller, On solutions of $\Delta$u = f(u), Comm.Pure Appl.Math. 10(1957), 503-510. https://doi.org/10.1002/cpa.3160100402
  3. R.Osserman,On the inequality $\Delta$u ${\geq}$ f(u), Pacific J.Math. 7(1957), 1641-1647. https://doi.org/10.2140/pjm.1957.7.1641
  4. C.Bandle, M.Marcus,Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J.Appl.Mathem. 58(1992), 9-24.
  5. C.Bandle, M.Marcus,On second order effects in the boundary behavior of large solutions of semilinear elliptic problems, Differential Integral Equ. 11(1998), 23-34.
  6. C.Bandle, M.Marcus,Asymptotic behavior of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary, Ann.Inst. Henri Poincare Anal Non Lineaire 12(1995), 155-171.
  7. J.Diaz,Nonlinear Partial Differential Equations and Free Boundaries, vol.I, Elliptic Equations, in: Pitman Research Notes in Mathematics Series, vol.106, Longman, 1985.
  8. A.V.Lair,a necessary and suffcient condition for existence of large solutions to semilinear elliptic equations, J.Math.Anal.Appl. 240(1999), 205-218. https://doi.org/10.1006/jmaa.1999.6609
  9. A. C. Lazer and P. J. McKenna,Asymptotic behavior of solutions of boundary blow-up problems. Differential integral equations 7(1994), 1001-1019.
  10. J.Matero,Quasilinear elliptic equations with boundary blow-up, J.Anal.Math. 69(1996), 229-247. https://doi.org/10.1007/BF02787108
  11. A. C. Lazer and P. J. McKenna, Asymptotic behavior of solutions of boundary blow up problems, Differential and Integral Equations 7(1994), 1001-1019.
  12. Claudia Anedda, Giovanni Porru,Boundary behavior for solutions of boundary blow-up problems in a borderline case, J.Math.Anal.Appl. 352(2009), 35-47. https://doi.org/10.1016/j.jmaa.2008.02.042
  13. F.Gladiali, G.Porru,Estimates for explosive solutions to p-laplace equations, in: Progress in Partial Differential Equations (Pont-a-Mousson 1997), vol.1, in: Pitman Research Notes in Mathematics Series, vol.383,Longman, 1998, 117-127.
  14. Z.Yang, Existence of positive entire solutions for singular and non-singular quasi-linear elliptic equation, J.Comm.Appl.Math, 197(2006), 355-364. https://doi.org/10.1016/j.cam.2005.08.027