Journal of applied mathematics & informatics

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

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE QUASILINEAR ELLIPTIC SYSTEMS WITH DIRICHLET BOUNDARY VALUE PROBLEMS

  • CUI, ZHOUJIN (Institute of Mathematics, School of Mathematics and Computer Science, Nanjing Normal University, Institute of Science, PLA University of Science and Technology) ;
  • YANG, ZUODONG (Institute of Mathematics, School of Mathematics and Computer Science, Nanjing Normal University, School of Zhongbei, Nanjing Normal University) ;
  • ZHANG, RUI (Department of Basic Education, Jinling Institute of Technology)
  • Published : 2010.01.30
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the system $$\{{{-{\Delta}_pu\;=\;{\lambda}f(\upsilon),\;\;\;x\;{\in}\;{\Omega}, \atop -{\Delta}_q{\upsilon}\;=\;{\mu}g(u),\;\;\;x\;{\in}\;{\Omega},} \atop u\;=\;\upsilon\;=\;0,\;\;\;x\;{\in}\;{\partial\Omega},}$$ where ${\Delta}_pu\;=\;div(|{\nabla}_u|^{p-2}{\nabla}_u)$, ${\Delta}_{q{\upsilon}}\;=\;div(|{\nabla}_{\upsilon}|^{q-2}{\nabla}_{\upsilon})$, p, $q\;{\geq}\;2$, $\Omega$ is a ball in $\mathbf{R}^N$ with a smooth boundary $\partial\Omega$, $N\;{\geq}\;1$, $\lambda$, $\mu$ are positive parameters, and f, g are smooth functions that are negative at the origin and f(x) ~ $x^m$ g(x) ~ $x^n$ for x large for some m, $n\;{\geq}\;0$ with mn < (p - 1)(q - 1). We establish the existence and uniqueness of positive radial solutions when the parameters $\lambda$ and $\mu$ are large.

Keywords

References

  1. D.D. Hai, R. Shivaji, Uniqueness of positive solutions for a class of semipositone elliptic systems, Nonlinear Anal., 66 (2007), 396-402. https://doi.org/10.1016/j.na.2005.11.034
  2. H.B. Keller, D.S. Cohen, Some positone problems suggested by nonlinear heat heat generation, J. Math. Mech., 16 (1967), 1361-1376.
  3. G. Astrita, G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, New York, 1974.
  4. L.K. Martinson, K.B. Pavlov, Unsteady shear flows of a conducting fluid with a rheological power law, Magnit. Gidrodinamika, 2 (1971), 50-58.
  5. A.S. Kalashnikov, On a nonlinear equation appearing in the theory of non-stationary filtration, Thudy. Sem. Petrovsk., 5 (1978), 60-68.
  6. J.R. Esteban, J.L. Vazquez, On the equation of turbulent filtration in one-dimensional porous media, Nonlinear Anal., 10 (1982), 1303-1325.
  7. L. Erbe, M. Tang, Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball, J. Differential Equations, 138 (1997), 351-379. https://doi.org/10.1006/jdeq.1997.3279
  8. M. Garcia-Huidobro, R. Manasevich, K. Schmitt, Positive radial solutions of quasilinear elliptic partial differential equations in a ball, Nonlinear Anal., 35 (1999), 175-190. https://doi.org/10.1016/S0362-546X(97)00613-5
  9. Z.M. Guo, 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.
  10. D.D. Hai, K. Schmitt, On radial solutions of quasilinear boundary value problems, Topics in Nonlinear Analysis, Progress in Nonlinear Differential Equations and their Applications, Birkhauser, Basel, 1999, pp.349-361.
  11. Z.M. Guo, Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations, Appl. Anal., 47 (1992), 173-190. https://doi.org/10.1080/00036819208840139
  12. E.N. Dancer, On the number of positive solutions of weakly nonlinear equations when a parameter is large, Proc. London Math. Soc., 53 (1986), 429-452. https://doi.org/10.1112/plms/s3-53.3.429
  13. S.S. Lin, On the number of positive solutions for nonlinear elliptic equations when a parameter is large, Nonlinear Anal., 16 (1991), 283-297. https://doi.org/10.1016/0362-546X(91)90229-T
  14. D.D. Hai, 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
  15. D.D. Hai, Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 52 (2003), 595-603. https://doi.org/10.1016/S0362-546X(02)00125-6
  16. L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations, 133 (1997), 179-202. https://doi.org/10.1006/jdeq.1996.3194
  17. A. Castro, M. Hassanpour, R. Shivaji, Uniqueness of nonnegative solutions for a semipositone problem with concave nonlinearity, Comm. Partial Differential Equations, 20 (1995), 1927-1936. https://doi.org/10.1080/03605309508821157
  18. P.L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. https://doi.org/10.1137/1024101
  19. H. Berestycki, L.A. Caffarelli, L. Nirenberg, Inequalities for second order elliptic equations with applications to unbounded domains, A Celebration of John F. Nash Jr., Duke Math J., 81 (1996), 467-494. https://doi.org/10.1215/S0012-7094-96-08117-X
  20. A. Castro, C. Maya, R. Shivaji, Positivity of nonnegative solutions for semipositone cooperative systems, Proc. Dyn. Syst. Appl., 3 (2001), 113-120.
  21. W.C. Troy, Symmetric properties in sνstems of semilinear elliptic equations, J. Differential Equations, 42 (1981) 400-413. https://doi.org/10.1016/0022-0396(81)90113-3
  22. D.D. Hai, R. Shivaji, An existence on positive solutions for a class of of semilinear elliptic systems, Proc. Roy. Soc. Edinburgh, 134A (2004), 137-141.
  23. Ali, A. Castro, R. Shivaji, Uniqueness and stability of nonnegative solutions for semipositone problems in a ball, Proc. Amer. Math. Soc., 117 (1993), 775-782. https://doi.org/10.1090/S0002-9939-1993-1116249-5
  24. Zuodong Yang, Qishao Lu, Nonexistence of positive radial solutions to a quasilinear elliptic system and blow-up estimates for Non-Newtonian filtration system, Applied Math. Letters, 16(4) (2003), 581-587. https://doi.org/10.1016/S0893-9659(03)00040-5
  25. G.B. Gustafson, K. Schmitt, Nonzero solutions of boundary value problems for secondorder ordinary and delay differential equations, J. Differential Equations, 12 (1972), 129-147. https://doi.org/10.1016/0022-0396(72)90009-5