Journal of applied mathematics & informatics

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

EXISTENCE OF BOUNDARY BLOW-UP SOLUTIONS FOR A CLASS OF QUASILINEAR ELLIPTIC SYSTEMS

  • Wu, Mingzhu (Colleg of Zhongbei, Nanjing Normal University) ;
  • Yang, Zuodong (Institute of Mathematics, School of Mathematics and Computer Science, Nanjing Normal University)
  • 발행 : 2009.09.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we consider the quasilinear elliptic system $\\div(|{\nabla}u|^{p-2}{\nabla}u)=u(a_1u^{m1}+b_1(x)u^m+{\delta}_1v^n),\;\\div(|{\nabla}_v|^{q-2}{\nabla}v)=v(a_2v^{r1}+b_2(x)v^r+{\delta}_2u^s)$, in $\Omega$ where m > $m_1$ > p-2, r > $r_1$ > q-, p, q $\geq$ 2, and ${\Omega}{\subset}R^N$ is a smooth bounded domain. By constructing certain super and subsolutions, we show the existence of positive blow-up solutions and give a global estimate.

키워드

참고문헌

  1. F. Cirstea and V. Radulescu,Entire solutions blowing up at infinity for semilinear elliptic systems, J. Math. Pures Appl. 81 (2002), 827-846.
  2. Ph. Clement, R. Manasevich and E. Mitidieri,Positive solutions for a quasilinear system via blow up, Comm. in Partial Diff. Eqns. 18, No.12 (1993), 2071-2106. https://doi.org/10.1080/03605309308821005
  3. Ph. Clement, D.G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. in P.D.E. 17, No.5/6 (1992), 923-940. https://doi.org/10.1080/03605309208820869
  4. Z.Zhang, Existence of large solutions for a semilinear elliptic problem via explosive subsupersolutions, Electrons.J.Differential Equations.2(2006),1-8.
  5. P. L. Felmer, R. Manasevich and F. de Thelin,Existence and uniqueness of positive solutions for certain quasilinear elliptic system, Comm.in P.D.E. 17 (1992), 2013-2029.
  6. Z. M. Guo, Existence of positive radial solutions for a class of quasilinear elliptic systems in annular domains, Chinese Journal of Contemporary Math. 17, No.4 (1996), 337-350
  7. LeiWei and MingxinWang, Existence and extimate of large solutions for an elliptic system, Nonlinear Analysis. in press
  8. A. V. Lair and A. W. Wood, Existence of entire large positive solutions of semilinear elliptic systems, J. Differential Eqns 164, No.2 (2000), 380-394. https://doi.org/10.1006/jdeq.2000.3768
  9. Huiling Li and MingxinWang, Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system, J. Differential Eqns 234,(2007), 246-266. https://doi.org/10.1016/j.jde.2006.11.003
  10. E. Mitidieri, G. Sweers and R. van der Vorst, Nonexistence theorems for systems of quasilinear partial differential equations, Differential Integral Equations 8 (1995), 1331-1354.
  11. E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic system in $R^N$, Diff. Integral Equations 9 (1996), 465-479.
  12. E. Mitidieri, A Rellich type identity and applications, Comm. in Partial Diff. Equations 18 (1993), 125-171. https://doi.org/10.1080/03605309308820923
  13. L. A. Peletier and R. van der Vorst, Existence and non-existence of positive solutions of non-linear elliptic systems and the biharmonic equations, Diff. Integral Eqns. 54 (1992), 747-767.
  14. J.Lopez-Gomez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations, 224(2006), 385-439. https://doi.org/10.1016/j.jde.2005.08.008
  15. J. Serrin and H. Zou, Non-existence of positive solutions of Lane Emden systems, Differential Integral Equations 9, No.4 (1996), 635-653.
  16. Z. D. Yang and Q. S. Lu, Non-existence of positive radial solutions for a class of quasilinear elliptic system, Comm. Nonlinear Sci. Numer. Simul. 5, No.4 (2000), 184-187. https://doi.org/10.1016/S1007-5704(00)90033-9
  17. Z.D. Yang, Existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems, J. Math. Anal Appl., 288(2003), 768-783 https://doi.org/10.1016/j.jmaa.2003.09.027
  18. Z.D. Yang and Q.S. Lu,Blow-up estimates for a quasilinear reaction-diffusion system, Math. Methods in the Appl. Sci., 26(2003), 1005-1023. https://doi.org/10.1002/mma.409
  19. Z.D. Yang and Q.S. Lu,Nonexistence of positive solutions to a quasilinear elliptic system and blow-up estimates for a quasilinear reaction-diffusion system, J. Computational and Appl. Math., 150(2003), 37-56. https://doi.org/10.1016/S0377-0427(02)00563-0
  20. J. Grrcia-Melian and J.D. Rossi,Boundary blow-up solutions to elliptic systems of competitive type, J. Differential Equations, 206(2004), 156-181. https://doi.org/10.1016/j.jde.2003.12.004
  21. D. Gibarg, N.S. Trudinger,Elliptic Partial Differential Equations of second Order, Springer, Berlin, 1983.
  22. G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12(1988), 1203-1219. https://doi.org/10.1016/0362-546X(88)90053-3
  23. Z. M. Guo and J. R. L. Webb, Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large. Proc .Roy. Soc. Edinburgh. 124A(1994), 189-198.
  24. J. Diaz, Nonlinear Partial Differential Equations and Free Boundaries, vol I, Elliptic Equations, in: Pitman Research Notes in Mathematics Series, vol. 106, Longman, 1985.
  25. C.V.Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992.
  26. J.Garcia-Melian, Nondegeneracy and uniqueness for boundary blow-up elliptic problems, J. Differential Equations, 223(2006), 208-227. https://doi.org/10.1016/j.jde.2005.05.001
  27. Z.D. Yang and Mingzhu Wu,Existence of boundary blow-up solutions for a class of quasilinear elliptic systems with subcritical case, Comm.Pure Appl. Anal., 6(2)(2007), 531-540.
  28. G.Diaz and R.Letelier, Explosive solutions of quasilinear elliptic equations:Existence and uniquness, Nonlinear Anal., 20(1993), 97-125. https://doi.org/10.1016/0362-546X(93)90012-H
  29. D.D.Hai and R.Shivaji, An existence result on positive solutions for a class of p-Laplacian systems, Nonlinear Anal., 56(2004), 1007-1010. https://doi.org/10.1016/j.na.2003.10.024
  30. Caisheng Chen, On positive weak solutions for a class of quasilinear elliptic systems, Nonlinear Anal., 62(2005), 751-756. https://doi.org/10.1016/j.na.2005.04.007
  31. M. Chhetri, D.D.Hai and R.Shivaji, On positive solutions for a class of p-Laplacian semipositone systems, Discrete Continuous Dyn. Syst., 9(4)(2003), 1063-1071.
  32. A.C.Lazer, P.J.McKenna, On a problem of Bieberbach and Rademacher, Nonlinear Anal. 21(1993), 327-335. https://doi.org/10.1016/0362-546X(93)90076-5
  33. O.A.Ladyzhenskaya, N.N. Ural'tseva,Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
  34. P.Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary point, Comm. Partial Differential Equations 8(7)(1983) 773-817. https://doi.org/10.1080/03605308308820285
  35. C.L.Liu, Z.D.Yang, Existence of large solutions for a quasilinear elliptic problem via explosive sub-supersolutions, Appl.Math.Comput 199(2008), 414-424. https://doi.org/10.1016/j.amc.2007.10.009
  36. C.L.Liu, Z.D.Yang, Boundary blow-up quasilinear elliptic problems of the Bieberbach type with nonlinear gradient terms, Nonlinear Anal. (in press).