East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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FUNDAMENTAL THEOREM OF UPPER AND LOWER SOLUTIONS FOR A CLASS OF SINGULAR (p1, p2)-LAPLACIAN SYSTEMS

  • XU, XIANGHUI (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY) ;
  • LEE, YONG-HOON (DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY)
  • Received : 2015.09.09
  • Accepted : 2015.09.16
  • Published : 2015.09.30
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Abstract

We introduce the fundamental theorem of upper and lower solutions for a class of singular ($p_1,\;p_2$)-Laplacian systems and give the proof by using the Schauder fixed point theorem. It will play an important role to study the existence of solutions.

Keywords

References

  1. R.P. Agarwal, H.S. Lu and D. O'Regan, Eigenvalues and the one-dimensional p-Laplacian, J. Math. Anal. Appl. 266 (2002) 383-400. https://doi.org/10.1006/jmaa.2001.7742
  2. X.Y. Cheng and H.S. Lu, Multiplicity of positive solutions for a $(p_1;\;p_2)$-Laplacian system and its applications, Nonlinear Anal. R.W.A. 13 (2012) 2375-2390. https://doi.org/10.1016/j.nonrwa.2012.02.004
  3. D.R. Dunninger and H.Y.Wang, Existence and multiplicity of positive solutions for elliptic systems, Nonlinear Anal. T.M.A. 29 (1997) 1051-1060. https://doi.org/10.1016/S0362-546X(96)00092-2
  4. D.R. Dunninger and H.Y. Wang, Multiplicity of positive radial solutions for an elliptic system on annulus, Nonlinear Anal. 42 (2000) 803-811. https://doi.org/10.1016/S0362-546X(99)00125-X
  5. A.G. Kartsatos, Advanced ordinary differential equations, Mancorp Publishing, Florida, 1993.
  6. R. Kajikiya, Y.H. Lee and I. Sim Bifurcation of sign-changing solutions for one-dimensional p-Laplacian with a strong singular weight; p-sublinear at $\infty$, Nonlinear Anal. 71 (2009), 1235-1249. https://doi.org/10.1016/j.na.2008.11.056
  7. Y.H. Lee, A multiplicity result of positive solutions for the generalized gelfand type singular boundary value problems, Nonlinear Anal. T.M.A. 30 (1997) 3829-3835. https://doi.org/10.1016/S0362-546X(97)00077-1
  8. Y.H. Lee, Eigenvalues of singular boundary value problems and existence results for positive radial solutions of semilinear elliptic problems in exterior domains, Diff. Integral Eqns 13 (2000) 631-648.
  9. Y.H. Lee, A multiplicity result of positive radial solutions for a multiparameter elliptic system on an exterior domain, Nonlinear Anal. 45 (2001) 597-611. https://doi.org/10.1016/S0362-546X(99)00410-1
  10. Y.H. Lee, Existence of multiple positive radial solutions for a semilinear elliptic system on an unbounded domain, Nonlinear Anal. 47 (2001) 3649-3660. https://doi.org/10.1016/S0362-546X(01)00485-0
  11. Y.H. Lee, Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus, J. Diff. Eqns 174 (2001) 420-441. https://doi.org/10.1006/jdeq.2000.3915
  12. Y.H. Lee and I. Sim, Existence results of sign-changing solutions for singular one-dimensional p-Laplacian problems, Nonlinear Anal. 68 (2008), 1195-1209. https://doi.org/10.1016/j.na.2006.12.015
  13. Y.H. Lee, I. Sim and X.H. Xu, An existence result for systems of vector p-Laplacian with singular weights, submitted.
  14. E.K. Lee and Y.H. Lee, A multiplicity result for generalized Laplacian systems with multiparameters, Nonlinear Anal. 71 (2009) e366-e376. https://doi.org/10.1016/j.na.2008.11.001
  15. E.K. Lee and Y.H. Lee, A global multiplicity result for two-point boundary value problems of p-Laplacian systems, Sci. China Math. 53 (2010) 967-984. https://doi.org/10.1007/s11425-010-0088-5
  16. I. Sim and Y.H. Lee, A new solution operator of one-dimensional p-Laplacian with a sign-changing weight and its application, Abstr. Appl. Anal. 2012 (2012), Article ID 243740, 1-15.
  17. X.H. Xu and Y.H. Lee, Some existence results of positive solutions for $\varphi$-Laplacian systems, Abstr. Appl. Anal. 2014 (2014) Article ID 814312, 1-11.

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