Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS INVOLVING GRUSHIN TYPE OPERATOR

  • Received : 2012.07.04
  • Published : 2014.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Using variational methods, we prove some results on the nonexistence and multiplicity of weak solutions for a class of semilinear elliptic systems of two equations involving Grushin type operators with sign-changing nonlinearities. We also shows that the similar results can be obtained for systems of m equations, where m is arbitrary.

Keywords

References

  1. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 4 (1973), 349-381.
  2. L. Boccardo and D. G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Differential Equations Appl. 9 (2002), no. 3, 309-323. https://doi.org/10.1007/s00030-002-8130-0
  3. J. F. Bonder, Multiple positive solutions for quasilinear elliptic problems with sign-changing nonlinearities, Abstr. Appl. Anal. 2004 (2004), no. 12, 1047-1056. https://doi.org/10.1155/S1085337504403078
  4. N. M. Chuong and T. D. Ke, Existence of solutions for a nonlinear degenerate elliptic system, Electron. J. Differential Equations 2004 (2004), no. 93, 1-15.
  5. N. M. Chuong and T. D. Ke, Existence results for a semilinear parametric problem with Grushin type operator, Electron. J. Differential Equations 2005 (2005), no. 107, 1-12.
  6. F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existstence (and non existstence) and symmetry of extremal functions, Comm. Pure Appl. Math. 54 (2001), no. 2, 229-258. https://doi.org/10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I
  7. N. T. Chung and H. Q. Toan, On a class of degenerate and singular elliptic systems in bounded domains, J. Math. Anal. Appl. 360 (2009), no. 2, 422-431. https://doi.org/10.1016/j.jmaa.2009.06.073
  8. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology I: Physical Origins and Classical Methods, Springer-Verlag, Berlin, 1985.
  9. V. Felli and M. Schneider, A note on regularity of solutions to degenerate elliptic equations of Caffarelli-Kohn-Nirenberg type, Adv. Nonlinear Stud. 3 (2003), no. 4, 431-443.
  10. V. V. Grushin, On a class of elliptic pseudo differential operators degenerate on a submanifold, Math. USSR Sbornik 13 (1971), 155-183. https://doi.org/10.1070/SM1971v013n02ABEH001033
  11. L. Iturriaga, Existence and multiplicity results for some quasilinear elliptic equation with weights, J. Math. Anal. Appl. 339 (2008), no. 2, 1084-1102. https://doi.org/10.1016/j.jmaa.2007.07.072
  12. T. D. Ke, Existence of non-negative solutions for a semilinear degenerate elliptic system, Proceedings of the international conference on Abstract and Applied Analysis (Hanoi Aug. 2002), World Scientific, July 2004.
  13. T. T. Khanh and N. M. Tri, On the analyticity of solutions to semilinear differential equations degenerated on a submanifold, J. Differential Equations 249 (2010), no. 10, 2440-2475 . https://doi.org/10.1016/j.jde.2010.06.002
  14. A. Kristaly and C. Varga, Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity, J. Math. Anal. Appl. 352 (2009), no. 1, 139-148. https://doi.org/10.1016/j.jmaa.2008.03.025
  15. F. Lascialfari and D. Pardo, Compact embedding of a degenerate Sobolev space and existence of entire solutions to a semilinear equation for a Grushin-type operator, Rend. Sem. Mat. Univ. Padova 107 (2002), 139-152.
  16. M. Mihailescu, Nonlinear eigenvalue problems for some degenerate elliptic operators on ${\mathbb{R}}^N$, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 3, 435-448.
  17. M. Mihailescu, G. Morosanu, and D. S. Dumitru, Equations involving a variable exponent Grushin-type operator, Nonlinearity 24 (2011), no. 10, 2663-2680. https://doi.org/10.1088/0951-7715/24/10/001
  18. M. Mihailescu and V. Radulescu, Ground state solutions of nonlinear singular Schrodinger equations with lack compactness, Math. Methods Appl. Sci. 26 (2003), no. 11, 897-906. https://doi.org/10.1002/mma.403
  19. M. K. V. Murthy and G. Stampachia, Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl. 80 (1968), 1-122. https://doi.org/10.1007/BF02413623
  20. K. Perera, Multiple positive solutions for a class of quasilinear elliptic boundary-value problems, Electron. J. Differential Equations 2003 (2003), no. 7, 1-5.
  21. V. Radulescu and D. Smets, Critical singular problems on infinite cones, Nonlinear Anal. 54 (2003), no. 6, 1153-1164. https://doi.org/10.1016/S0362-546X(03)00131-7
  22. N. T. C. Thuy and N. M. Tri, Some existence and non-existence results for boundary value problems (BVP) for semilinear elliptic degenerate operators, Russ. J. Math. Phys. 9 (2002), no. 3, 366-371.
  23. N. T. C. Thuy and N. M. Tri, Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations, Nonlinear Differential Equations Appl. 19 (2012), no. 3, 279-298. https://doi.org/10.1007/s00030-011-0128-z
  24. N. M. Tri, On Grushin's Equation, Matemachicheskie Zametki 63 (1988), 95-105.
  25. N. M. Tri, Critical Sobolev exponent for degenerate elliptic operators, Acta Math. Vietnamica 23 (1998), no. 1, 83-94.
  26. B. J. Xuan, The eigenvalue problem of a singular quasilinear elliptic equation, Electron. J. Differential Equations 2004 (2004), no. 16, 1-11.